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UNIVERSITY 
OF FLORIDA 
LIBRARIES 




COLLEGE LIBRARY 



BUDGET CONTROL 
AXD COST BEHAVIOR 



1959 Award Winner 
THE FORD FOUNDATION DOCTORAL DISSERTATION SERIES 



A dissertation submitted in partial 
fulfillment of the requirements for 
the degree of Doctor of Philosophy 
at Carnegie Institute of Technology 



BUDGET CONTROL 



AND COST BEHAVIOR 



ANDREW C. STEDRY 



PRENTICE 



1960 
HALL, I IS C 

Englewood Cliffs, N. J. 



© — 1960 by PRENTICE-HALL, INC. 

Englewood Cliffs, N. J. 

All rights reserved. No part of this book 

may be reproduced in any form, by mimeograph or any other means, 

without permission in writing from the publishers. 

L. C. Catalog Card Number: 60-11587 



• Printed in The United States of A merica 

08522 — C 



Foreword 



This volume is one of five doctoral dissertations selected for publication 
in the first annual Doctoral Dissertation Competition sponsored by the 
Program in Economic Development and Administration of The Ford 
Foundation. The winning dissertations were completed during the 
academic year 1958-59 by doctoral candidates in business administration 
and doctoral candidates in the social sciences and other fields relevant 
to the study of problems of business. 

The dissertation competition is intended to generalize standards of 
excellence in research on business by graduate students. It should give 
widespread professional recognition to persons recently awarded doc- 
torates in business whose dissertation research is especially distinguished 
by its analytical content and strong roots in underlying disciplines. It is 
also intended to give recognition to a select number of persons outside 
business schools who in their doctoral dissertations pursued with dis- 
tinction interests relevant to business. 

The dissertations selected include, in addition to Dr. Stedry's 
monograph : 

Computer Models of the Shoe, Leather, Hide Sequence 
Kalman J. Cohen 

Graduate School of Industrial Administration 
Carnegie Institute of Technology 

Poly a Type Distributions in Renewal Theory, with an Application to an 
Inventory Problem 

Frank Proschan 

Department of Statistics 

Stanford University 

The Structure of a Retail Market and the Market Behavior of Retail Units 
Bob R. Holdren 
Department of Economics 
Yale University 

Some Personality Determinants of the Effects of Participation 
Victor H. Vroom 
Department of Psychology 
University of Michigan 



vi Foreword 

The many high-quality dissertations submitted were judged by the 
most exacting professional standards. Specific criteria included: 

a. Importance of the problem and originahty of approach; 

b. Use of the most appropriate and powerful tools of analysis; 

c. Clear relation to the relevant theoretical framework or a contri- 
bution to theory; 

d. Direct relevance to the practice of business or management; 

e. Clarity and effectiveness of presentation. 

An examination of all five volumes in this series will reveal that four 
of the five make considerable use of mathematical and statistical tools. 
This reflects the increasing importance of modern quantitative methods 
in the study of business. On the other hand, the use of quantitative 
techniques should certainly not be considered a sine qua non of rigorous 
research in business. It is hoped that in future years it will be possible to 
select for publication a greater number of nonmathematical dissertations 
of the highest quality. 

On behalf of The Ford Foundation, I wish to express my sincere 
appreciation to the Editorial Committee for its painstaking effort in 
selecting the winning dissertations. The scholars who served as members 
of the Committee for the first year's competition were Robert Ferber, 
Research Professor of Economics, University of Illinois; Sherman J. 
Maisel, Professor of Business Administration, University of California 
(Berkeley); and William Foote Whyte, Professor, New York State 
School of Industrial and Labor Relations, Cornell University. 

The work of the Editorial Committee was materially aided by a group 
of six readers, who spent hundreds of hours in conscientious examination 
of the dissertations submitted. The Foundation and the Committee wish 
to thank Professors Austin C. Hoggatt and Julius Margolis of the Uni- 
versity of Cahfornia (Berkeley), Henry A. Landsberger and Seymour 
Smidt of Cornell University, and Vernon K. Zimmerman and Thomas A. 
Yancey of the University of Illinois for their service as readers m the first 
year of the competition. 

Finally, my colleagues and I wish to acknowledge the substantial 
contribution of Prentice-Hall, Inc., to the pubUcation and distribution 
of the selected dissertations. 

Thomas H. Carroll 

VICE PRESIDENT 
THE FORD FOUNDATION 

New York, New York 
January, 1960 



Preface 



This study constitutes an attempt to bring to bear on a particular man- 
agement problem — the problem of budgetary control — knowledge 
which is drawn from five disciplines: economics, psychology, organ- 
ization theory, mathematics, and accounting. 

It is difficult in a document of limited size to provide in every instance 
a comparison of the opposing or reinforcing views of the main bodies of 
theory in each of the disciplines. Furthermore, as is well known, each 
discipline has its own peculiar manner of expression. Thus, if it would 
appear at various points in this document that some lack of consistency 
in terminology is present, it should be understood that comparisons must 
often be drawn with terminology that is alien to one discipline or an- 
other. 

The aim of this study is to explore various approaches to the problem 
of budget control, using tools supplied by the various disciplines where it 
is felt they are applicable. It might be said that the approach was thus 
''problem-oriented" rather than ''tool-oriented." The three approaches 
that will be presented are (1) a mathematical systems model of individual 
behavior in a goal- or budget-striving situation, (2) an empirical investi- 
gation of the performance of individuals under varying budget condi- 
tions, and (3) the linear programming formulation which attempts to 
deal with some of the planning and coordinating aspects of budgeting. 
The means used for this attempt to at least provide a step in the direction 
of a scientific basis for budget control are sufficiently diverse in method 
that their approach to the same end is frequently not obvious. However, 
it would seem most unwise to attempt to solidify, at this stage of the 
development, any monistic hne of inquiry into a subject which, albeit so 
old in the annals of history, is young in the annals of science. 

ACKNOWLEDGMENTS 

The work carried on in the thesis was in part supported by the Office of 
Naval Research to which I owe my thanks. Parts of the dissertation 
appear in more or less complete form as Office of Naval Research 
memoranda numbers 60, 61, and 63^ To the administrator of the proj- 

iReferences (75), (76), (77) of this thesis. 



viii Preface 

ect, Professor Peter R. Winters, and the director, Professor G. L. Bach, 
Dean of the Graduate School of Industrial Administration, Carnegie 
Institute of Technology, go my thanks for their aid in procuring the 
necessary funds for carrying out the empirical investigation. 

I owe my thanks to the Graduate School of Industrial Administration 
in whose name fellowships were granted for the years 1957-59, during 
which time the research reported in this paper took place. 

It would be virtually an impossible task to credit all of the people 
who have contributed to my academic background, and thus to the 
pursuit of the degree for which this paper is presented. I will, however, 
attempt to mention those who were most inspiring and helpful. 

I have benefited greatly in the past from association with Professor 
Franco Modighani whose lectures comprised the major part of my formal 
instruction in economic theory and with whose aid I have been able to 
pubhsh my first journal article.^ The idea upon which Chapter 2 is based 
was discussed with Professor Herbert A. Simon two years ago, and from 
this discussion, with Professor Simon's encouragement, the relationships 
between budgets, aspirations, and performance became the nucleus of 
the document. 

The knowledge of psychology necessary to carry out this study was 
obtained with the aid of Professors Harold J. Leavitt, Walter Reitman, 
and Richard Willis, who directed me toward appropriate works in the 
field and discussed with me the plans for the empirical research to be 
reported. 

I am deeply indebted to my committee. Professors William W. 
Cooper, Merton H. Miller, Richard M. Cyert, and Harold Leavitt for 
their full cooperation during the progress of the work. Their criticisms 
were constructive and helpful. 

The experiment was performed with the aid of my colleague, ]\Ir. 
Charles P. Bonini and Messrs. James Lemon, and Richard Young, 
students in the Graduate School of Industrial Administration. 

I am indebted to Miss Sandra Kinney who typed the majority of the 
document, who faithfully devoted every night, and weekends as well for 
several weeks, in order to ensure the timely presentation of the document. 
Mrs. Joan Anderson came to my rescue at the eleventh hour when it 
seemed, in spite of all efforts, that the typing of the document could not 
be completed. My thanks go to Mrs. Rita Carlson and Miss Dolores 
Miller who also shared in the preparation of the document. 

There are, however, two people without whose efforts the thesis would 

2"A Note on Interest Rate and the Demand for Money," Review of Economics 
and Statistics, August, 1958. 



Preface ix 

never have been written. Professor William W. Cooper provided the 
initial stimulus which caused me to seek a research degree, and since that 
time he has been unfailingly devoted to my progress toward that degree, 
and toward the research career beyond it. His aid, encouragement, and 
patience up to the hour that the document was delivered could not have 
been exceeded. 

Pat, my wife, has given me the support and understanding that 
enabled me to carry through my intention to complete the dissertation. 
She worked with me throughout the long nights which accompanied the 
final stages of the preparation of this thesis. 

Andrew C. Stedry 



«■ 



Contents 



CHAPTER 1 

PRELIMINARY CONSIDERATIONS 1 



1.1. The purpose of control 1 

1.2. Budgets: definition and scope 3 

1.3. Budgets: standards and standard costs 5 

1.4. How a budget controls 9 

1.5. Some remarks on budget control and economics 13 



CHAPTER 2 

A MATHEMATICAL MODEL OF A BUDGET 

CONTROL SYSTEM 17 



2.1. Introduction 1 7 

2.2. The budget as a determining factor in formation of 

aspiration levels 19 

2.3. Existing models of aspiration level determination 19 

2.4. Scope and postulates of the model 23 

2.5. Mathematical formulation of the model 26 

2.6. Behavior of the system in response to a specific rate of 

budget reduction 2S 

2.7. The oscillatory system 34 

2.8. A technological constraint 3S 

2.9. Conclusions 40 



XI 



Xll Contents 

CHAPTER 3 

AVAILABLE EMPIRICAL INFORMATION ^3 

3.1. Introduction J^ 

3.2. Laboratory experiments on the level of aspiration 43 

3.3. Experiments on the behavior of animals 50 

3.4. Studies of utility measurement 62 

3.5. Some field studies and a "practicar' example 56 

3.6. Conclusion 59 



CHAPTER 4 

AN EXPERIMENT 61 



4.1. Introduction 61 

4.2. The experimental task 63 

4.3. The reward structure and budget 67 

4.4. Aspiration level determination 71 

4.5. Experimental procedure 74 

4.6. Structure of the design and subjects utilized 74 

4.7. Analysis of results — performances 76 

4.8. Analysis of results — aspiration level 81 

4.9. Conclusions 89 



APPENDIX 4A 

DETAILS OF THE EXPERIMENTAL DESIGN 93 



4A.1. Task selection 93 

4A.2. Sample documents 95 



APPENDIX 4B 

ADDITIONAL STATISTICAL RESULTS 107 



Contents xiii 

CHAPTER 5 

A MATHEMATICAL MODEL FOR 

BUDGETARY PLANNING 113 

5.1. Introduction 113 

5.2. The hierarchy of the factors of production 115 

5.3. The concept of limited substitution — a definition 116 

5 .4. Form of the production function under limited substitution 117 

5.5. Example of a limited substitution problem 121 

5.6. Computation scheme format 127 

5.7. Optimization procedure 127 

5.8. Power of the model 136 

5.9. Parametric programming of the model 139 

5.10. Extension of the model to the next higher level 

in the hierarchy 1^0 

5.11. Summary I4I 



CHAPTER 6 

SUMMARY AND DIRECTIONS FOR 

FURTHER RESEARCH lU 



6.1. Introduction I44. 

6.2. Planning and control 1^5 

6.3. Some selected quotations 149 

6.4. Some directions for further research 154 

BIBLIOGRAPHY 155 



CHAPTER 1 



Preliminary Considerations 



1.1. The Purpose of Control 

This is a study in cost control which has as its referent the context 
of an individual firm where cost reports, budgets, standards, and like 
mechanics are generally employed to influence cost behavior.^ It does 
not deal — or at least it does not deal directly — with those aspects of 
cost calculation and planning as they bear on problems such as pricing, 
asset acquisition (and disposition), etc. Because much of the literature 
in accounting (and practically all of the literature in economics) ^ is 
devoted to one or the other of these two subjects, it will be useful to try 
to place this study in the perspective that may be secured by reference 
to a few pertinent quotations. 

Consider, for instance, the following statement as quoted from a 
report of the National Association of Accountants:^ 

Cost control has as its objective production of the required quality 
at the lowest possible cost attainable under existing conditions. 

Contained in this definition are terms such as ''lowest possible,'* 



^Specifically, other managed entities such as government agencies, labor unions, 
etc., are not considered as such. The kind of enterprise which the author has tried 
to bear in mind is one which is exemplified by a manufacturing firm of the kind which 
is believed to be typical in America. 

2A standard reference which is said to go far towards synthesizing economics and 
accounting is J. M, Clark (19). It will perhaps emphasize what is at issue here by 
stating that this book has almost nothing to say on those aspects of accounting with 
which this thesis is concerned. 

^Formerly the National Association of Cost Accountants. (73a), p. 443. 



2 Preliminary Considerations 

"attainable" and "existing conditions." Although of general applica- 
bility, these statements need to be sharpened to render them pertinent 
here. It is now known, for instance, that under a great variety of 
situations it is possible to plan production mthout any detailed knowl- 
edge of the cost function; existing conditions can be assumed.^ Does 
this cost-independent planning ability render the numerous and various 
kinds of activities that cost accountants normally undertake superfluous 
insofar as they emanate in cost reports where (numerical) data are 
presented to those whose primary duty it is to keep costs as low as 
possible? Under the usual existing conditions — or at least those usually 
posited in economic theory ^ — this is the case provided that, among other 
things, it is assumed that plans are carried forward to their fruition in 
actual operations. On the other hand, if control is the issue so that a 
mechanism, and especially a mechanism involving human intermedi- 
aries, must be considered as the normal way that plans are (possibly) 
translated into action, then equally severe questions arise as to the 
standard emphases on "accurate" and timely reports which are art to 
a common cloth for all recipients of this information. 

The usual cost or budget report tends to be a statement of goals in 
terms of levels; e.g., a standard cost is prescribed, the "actual" level 
is recorded and variances between the two (possibly) noted. Some 
consequences of this kind of reporting will be traced in later portions of 
this thesis (e.g.. Chapter 5) and will be shown to give "unwanted" 
results even when the ideal assumption of exact correspondence between 
plans and operations can be made. In the earlier portion of this thesis, 
and in contrast to most of the existing literature, the emphasis will be 
on rates (or variations between levels in succeeding periods). 

In this context the objective of budget control will be given an 
interpretation which is sharper than the one cited above. ^ As it will be 
treated here, the objective of budget control is to increase long-run profit 
at the fastest possible rate; or alternatively, at a given output, to reduce costs 
at the fastest possible rate. For the moment, at least, the question of 
just what rate is the "fastest possible" will be overlooked in order to 
devote attention to the definition and discussion of the implements of 
control, the relationship of budget control to the theory of the firm, 
and a brief description of the development to follow. 



iCf. F. Modigliani and F. Hohn (62). See also A. Charnes, W. W. Cooper, and 
B. Mellon (17) and their earlier paper (11). 
^Specifically, diminishing returns to scale. 
^It may not even be in accord with what the authors of that report had in mind. 



Preliminary Considerations 3 

1.2. Budgets: Definition and Scope 

In an attempt to eradicate, or at least mitigate, some of the ambiguity 
which will result from the particular usage of the term ''budget" in this 
thesis, it is necessary to relate it to the definitions in common use. The 
most comprehensive use of the term is exemplified by the following 
definition of ''budget" by Eric Kohler as:^ 

1. A financial plan serving as a pattern for and a control over future 
operations; 

2. hence, any estimate of future costs; 

3. a systematic plan for the utilization of man power, material or other 
resources. 

Implicit in Kohler's definition is the existence of a multiplicity of 
purposes for which budgets are constructed. 

Two major functions are, however, immediately discernable. First, 
a budget may serve as a plan, indicating requirements of certain factors 
(e.g., cash, productive capacity) at some future date which serves the 
function of providing information for subsequent decisions and possibly 
guiding them. Second, a budget may serve as a control, containing 
criteria of cost or performance which will be compared with actual data 
on operations, thus facilitating evaluations and possible encouraging or 
even enforcing some measure of efficiency. 

As may be already apparent, these separate functions (i.e., planning 
and controlling) need not be mutually exclusive nor, in practice, is it 
unusual for both to be represented in a single document. That these 
functions are (rightly or wrongly) fused is aptly indicated by the fol- 
lowing description of "production planning and control" by MacDonald :^ 

. . . one of the essential steps in the preparation of the production budget 
is the translation of sales estimates into specific production plans. 
While this activity is primarily the responsibility of the production 
executive, usually exercised through the head of planning or a produc- 
tion control department, it is so fundamental to practical budgetary 
control that it is essential that the budget executive at least be familiar 
with the essential features of it. 

There is certainly no doubt indicated in MacDonald's remarks about 
the advisability, or even necessity, of interlacing the planning and 



1(44), p. 67. 
2(53), p. 101. 



4 Preliminary Considerations 

control functions to the point where they become indistinguishable. A 
question might be raised, however, as to whether the interrelationship 
described can, in fact, be achieved with only one set of budgeted figures 
— a set which would need to serve both planning and control functions 
at various tiers (and over various persons) in an organization. Consider, 
for instance, the impact of the following remark as quoted by two other 
authors.^ ''A good plan (e.g., a budget or sales forecast) does not 
necessarily yield a good control.'' Also, "Good planning data and good 
control data are not necessarily the same."^ Therefore, it is e^ddent that 
there is some room for disagreement as well as some need for clarification 
in these areas, planning and control. 

In order to clarify this distinction, reference will be made to sales 
budgets where it is usual for distinctions of this kind to be recognized in 
the literature,^ possibly because widespread divergencies between plans 
and actual operations are more frequent than in, say, production or 
financial budgeting. One type of sales budget (frequently termed a 
"quota") is designed specifically as a control device. Its aims are to 
effect the motivation and guide the judgment of the salesmen b\' com- 
parison of budgeted and actual performance. This comparison may 
(and sometimes is) re-enforced by connection with various rewards and 
penalties. On the other hand, the tie between these quotas and the 
planning of output is often extremely loose. The planned output is often 
based on "estimated" or "expected" sales, and the relations of these 
expectations to the quotas suggests an assumption that at least some 
of the quotas will not be achieved.^ A question arises as to wh}" the 
"quota" concept is generally not carried over into other areas of bud- 
geting — e.g., production — as a control device. As far as may be 
discerned, the reasoning is somewhat as follows: the budget must ser^^e 
as a coordination device. Hence production must be planned so that 
the needs emanating from "expected sales" wiU be met along ^^ith other 
criteria, such as the size or fluctuations in inventory, that are regarded 
as prudent. The assumption which is made in practice (or at least in 
descriptions of practice) is that the figures to be used for control purposes 



lA. Charnes and W. W. Cooper (11). 

mid. 

^See, for example, Heckert (34), Chapter 11. 

^A third figure is sometimes apparent. A sales "forecast" emanating from the 
sales department may be adjusted downward (to compensate for anticipated opti- 
mistic bias) to obtain the sales expectation. 



Preliminary Considerations 5 

and the estimate of needs (i.e., the production plan) are the same.^ It is 
an hypothesis of this thesis that the equahty of the figures used for 
the control and planning budgets need not be assumed, but that its 
desirability is a testable proposition. Or, in other words, does some 
figure other than the planned amount, when used in the control budget, 
produce a performance which is actually closer to the planned amount? 
The questions which arise regarding the disparity of plans and con- 
trols indicate that the "budget process" is actually not a homogeneous 
mechanism but rather a collection of processes with a variety of aims 
and procedures of application. The principles of budgetary practice to 
be investigated in Chapters 2, 3, and 4 of this thesis concern only the 
control aspects. It may be assumed throughout these chapters that 
a "budget" can be interpreted as a "control budget" unless otherwise 
specified. The planning and forecasting aspects of budgeting treated 
in the thesis are largely contained in Chapter 5 although, as will become 
evident, the emphasis is on the types of planning considered necessary 
for adequate control. 

1.3. Budgets: Standards and Standard Costs 

In order to convey the applicability of the treatment of "budgets" 
in this paper to control systems using "standards" and "standard costs" 
as elements of control, the similarity of these various elements will now 
be discussed. It will first be desirable to examine "standards" and 
"standard costs" to provide a framework for the discussion of similarities 
and differences. 

Both "standards" and "standard costs" are so intimately related 



^This assumption is typified by some remarks of Rautenstrauch and Villers (64). 
They state: 

The yearly production budget is not equal to the sales forecast, nor to the sales 
forecast less inventory on hand but to sales forecasts plus {or minus) the increase 
(or reduction) of inventory required to bring the actual inventory to the level of budgeted 
inventory (p. 114). 

This budget is the only one which they propose for control purposes. Note that 
the distinction between what the budget is not and what it is is only one which 
(algebraically) assures that the estimates provide for a continuing enterprise. The 
possibility that estimated need and the need as stated in the control budget may not 
agree is not considered. It should be noted in this connection that Professors 
Rautenstrauch and Villers are industrial engineers. Their views, however, do not 
differ from those indicated in the citation of Mr. MacDonald (53), an executive, 
nor markedly from those of Professor Heckert (34) (Chapter 18), an accountant, 
on the subject of production budgets. 



6 Preliminary Considerations 

that the one is generally included with the other in any description of 
cost or profit control mechanisms. It is thus necessary, in order to 
maintain contact with the main body of accounting literature, to explain 
the context in which they (i.e., ''standards" and "standard costs") will 
be used in this thesis. 

In order to introduce these concepts as they will be used here, the 
definitions of Eric Kohler will be cited. He defines a standard as, "A 
desired attainment; a performance goal; a model. "^ It will be noted that 
Kohler views a standard as something to be striven for, and that al- 
though various types of schemes are used to set standards, the basic 
correspondence between standard and goal remains unaltered. Stand- 
ards are frequently encountered in such specific contexts as ''standard 
time," "standard material usage," etc. In contrast to standard, Kohler 
defines standard cost as, "A forecast or predetermination of what costs 
should be under projected conditions, serving as a basis for cost control, 
and as a measure of productive efficiency when ultimately compared 
with actual costs. "^ 

Even with the fairly explicit definitions of Kohler, the problems of 
classification are not always straightforward. For example, items such as 



1(44), p. 389. 

^Ibid. It should be noted that there are several types of standard costs in common 
use. Since these distinctions are not of prime interest here they will be discussed 
only brief! j\ A first classification may be made into two tj'pes: basic standards and 
current standards. The basic standard is essentially an index number. It is not used 
for control purposes, other than to exhibit a trend, and frequently is an old standard 
or actual cost as of a given date, etc. On the other hand, current standards represent 
the type used in the ordinary context of standards for cost control. These may be set 
in a variety of ways and may be further subdivided on this basis into two classifica- 
tions: estimates and standards. Both are expected to relate to current and future 
production and "the difference between the two is even conceptually a matter of 
degree. Estimated costs are the looser of the two." (A paraphrase of W. W. Cooper 
(21), Chapter V, pp. 18-19.) 

Additional distinctions, often made, are ideal (or perfection) standards and attain- 
able standards. These are both current standards, but the attainable standards are 
assumed to be able to be obtained under conditions of a reasonable degree of efficiency 
and effort. They may be either estimates based on past performance or engineering 
standards. The engineering standards are set by time study for labor cost or similar 
devices for other types of cost. These, too, are in reality estimates, since the standards 
are often subject to negotiation and there exists no reliable scientific basis upon which 
to justify an assumption of precision. So-called ideal standards are generally engi- 
neering standards, intended to describe the cost that could be attained under "opti- 
mum" efficiency. These are the closest to the concept of optimal cost in economic 
terms, but generally speaking each standard is set individually so that the factor 
interactions assumed in economic theory are not considered. 



Preliminary Considerations 7 

"standard labor costs" and "standard material costs" can be classified 
as "standard costs" and passed by without further discussion, since an 
absence of ambiguity in these classifications is usually assumed. But 
what about "standard overhead expense" and like items? Are these a 
"standard," or a "standard cost"? It is evident that there is some 
difference in the dimensions, at least as far as control is concerned. 
Whereas goals or levels used for control which are expressed in units 
other than dollars are necessarily classed as standards (or budgets, as 
will be explained later), those which are expressed in dollar terms are 
frequently classed as standard costs, but not invariably. 

In practice, moreover, a "standard time" (expressed in hours) for 
a particular operation extended by a "standard rate" (cost per hour) for 
a worker employed on the operation is usually termed a "standard labor 
cost" for the operation. Any two out of the three figures in this case 
may serve as the basis for control; the particular pair chosen is a matter 
of convenience. Conceptually, there is little difference between con- 
trolling input or output in physical units as opposed to dollar terms, 
although they may require different procedures. ^ 

From the above comments it may be inferred that the choice of a 
common denominator — a numeraire — is in reality fairly arbitrary, and 
hence the terms "standard" and "standard cost" can be used more or 
less interchangeably, at least in a theoretical document of this kind. 

Another issue which requires clarification involves the difference 
between budgets and standards. The distinctions made in the hterature 
vary considerably from author to author so that a concise summary is 
difficult to achieve. A view which is widely held, however, is that the 
difference is one of scope. This viewpoint is exemplified in the following 
remarks of S. Henrici:^ 

. . . Budgets are customarily set for all departments in the company, 
from sales to manufacturing. But standards are frequently set only for 
the manufacturing divisions and can, indeed, be confined to controllable 
costs in a limited number of cost centers . . . 



^Problems exist, of course, which may definitely indicate preference of one type 
of unit or the other. In the case of nondivisable joint costs, for example, dollar 
figures may be misleading. On the other hand, in the control of operations which 
involve large-scale aggregations of items with a multiplicity of physical units, dollar 
amounts may provide the only practicable solution. 

2(35), p. 232. A quotation from this same author, to be given shortly, indicates, 
however, that issues of purpose may also be used to distinguish between "budgeted" 
and "standard" cost. 



8 Preliminary Considerations 

Similar statements may be found in Lang, ^IcFarland, and Schiff 
(-46), and Heckert (34), who consider scope the essential difference. 
Henrici, however, considers scope only one of the distinctions, and not 

the primary one. He states: 

The first distinction between standards and budgets is one of 
purpose. Budgets are statements of expected [sic] cost . . . Standards 
on the other hand, do not necessarily show what costs may be expected 
to be [sic] but rather what they might be if certain highly desirable 
performances are attained . . .^ 

On the other hand, I. Wayne Keller proposes a distinction which 
might well be considered dictated by (in the linguistic sense) "common 
usage." He writes:^ 

... for control purposes the terms "standard" or "standard cost" are 
applied to the measurement and control of the costs of direct material, 
direct labor, and scrap [sic]. Expense is controlled through expense 
"budgets" rather than expense "standards." 

It is apparent from the foregoing conflicting distinctions that no 
common denominator exists upon which to base a single, well-defined 
criterion of separation between standards (including standard costs) and 
budgets. The situation is perhaps best explained by the foUo'^dng 
statement of the National Association of Accountants' (formerly 
N.A.C.A.) report, "A Re-Examination of Standard Costs." In relating 
standard costs to the "scientific management" movement, they note:^ 

Historically, standard costs as we now know them and business 
budgeting developed at about the same time, but in the earlier years 
their development was largeh' separate. Standard costs developed in 
the factory while budgeting was applied first to the financial aspect of 
business. Later on it was realised [sic] that both were merely applica- 
tions of the same management philosophy and that they were comple- 
mentary parts of a complete programme of cost control. 

This view, which minimizes the difference between budgets and 
standards, would seem to be most sensible in light of the profusion of 
conflicting statements which may be found. It also seems, however. 



^Ibid. It should be noted that this distinction is not consistent with the definition 
of Kohler, supra, which is used in this paper. (Henrici fists two other distinctions 
which also depend upon a definition of budget at variance with Kohler's.) 

2(40), p. 97. 

3(73a), p. 438. 



Preliminary Considerations 9 

that planning and forecasting budgets might properly be differentiated 
from standards, although standards are frequently used in the deter- 
mination of the plans and forecasts. 

On the other hand, a control budget, as defined supra, carries with it 
the connotation of a "goal" or "desired attainment" which is noted in 
Kohler's definition of standard. It may thus be seen that "budgeted 
performance" and "standard performance" differ only in name if they 
are both goals or desired attainments. 

Agreement among writers is more or less general only in the matter 
of the difference in scope exhibited by budgets and standards. It would 
not be contrary to this consensus if some distinction were made; e.g., a 
budget is a goal on a large scale, a standard a goal on a smaller scale. 
But even this distinction would appear to be artificial from the stand- 
point of classification by function since both a budget and a standard 
may, in this context, serve the same purpose. 

In any case, in this thesis, the meaning of budget (as a control 
device) will be interpreted as a goal or desired attainment, and the 
foregoing discussion and quotations used should suffice to justify, to a 
first approximation, why findings from a study of "budgets" should 
also be applicable to "standards" or "standard costs" as one or the 
other (or all three) are used as part of a cost control system. 

1.4. How a Budget Controls 

In the preceding sections of this chapter, it was noted that there is 
a control aspect of budgeting that is distinguishable, in some sense, 
from either planning or forecasting. Assume for the moment that this 
distinction is valid and can be sufficiently well demarcated; let it then 
be assumed that a mechanism has been created for the sole purpose of 
producing and administering a control budget. A question may then 
be asked as to just what the budget control system and the budget 
documents (which are an integral part of the system) should consist of 
in order to insure that the cost or performance elements budgeted are 
in fact being controlled. 

It would seem reasonable that in order to insure some form of control, 
the process by which control is exercised should be analyzed. In other 
words, assurance of control would seem to require some answer to the 
question, "Just how do budgets control?" This issue is rarely addressed 
in the budgeting literature except by implication and by reference to 
"experience," "practice," and intuitive appeals that are more or less 
plausible (when stated). 



10 Preliminary Considerations 

A more than usually lucid treatment is presented by Henrici. He 
notes :^ 

The difference in a given period between actual cost and standard 
cost, known as the "variance," tells management to what extent costs 
can be controlled. The variance itself is not a control, for costs are not 
controlled by compiling statistics about them. The control consists of 
the steps that management takes to regulate or limit costs. And the 
effectiveness of these steps is gauged by the degree to which actual costs 
approach standard; in other words, by the size of the variance. 

An important feature to be noted in Henrici's remarks is the absence 
of the commonly held assumption that the means of reporting and 
controlling are the same.^ A second unusual feature is the concept of an 
approach to standard as a criterion of effectiveness. It should be noted 
in this regard that Henrici's definition of standard involves a concept 
not far removed from the ''technological optimum" of economics. He 
considers standards as emphasizing ''what should be" and having a 
"primary purpose of establishing a 'sea level/ so to speak, from which 
to measure cost altitudes."^ However, standards are frequently set by 



1(35), p. 154. 

2Cf., for example, Lang, McFarland, and Schiif (46), who state, "Control implies 
the desired objectives through the measurement of results, especially through com- 
parative reports." (p. 435.) Keller (40) likewise appears to overestimate the role of 
the accounting function. He notes: 

The first requisite for the control of material costs is organization, with 
responsibilities clearly established for all phases of the control problems. The 
accountant is the keystone in such an organization, for control will be no better 
than the accounting records and data which are established, (p. 158) 
Both of these authors depend upon the efficiency of their reporting schemes for 
control. It would seem, however, as though the best reporting scheme would be 
totally impotent as a control if there were no mechanism for translating reports into 
action. A reductio ad absurdum is sufficient to demonstrate the fallacy in both 
statements. If both supervisors and budgeted personnel were to ignore all reports — 
a possibility not excluded by the authors' statements — they would be valueless, 
regardless of how "good" they were as reports. 

3(35), p. 5. But note, however, that there is, within these broad directives, a 
problem of measurement of standards. Even in the area in which the most extensive 
Work on standards has been performed — the calculation of standard time — the 
issues are not clear-cut. March and Simon (57) point out that: 

. . . Often it is unclear whether standard times reflect "average time using 
average skill and average effort," "minimum time" or "average time over a series 
of trials by individuals randomly selected from a pool of industrial workers." 
(p. 16) 

It is thus apparent that, even with the best of intentions, a standard can be 
misleading. 



Preliminary Considerations 11 

a criterion which is at best awkwardly paraphrased somewhat as follows: 
''Standards should be set so that they are 'attainable but not too 
loose'. "^ If standards are interpreted in terms of this latter criterion, 
an "approach to standard" implies little more than an approach to a 
level of performance which was a priori assumed to be approachable, 
or perhaps more important, capable of betterment. 

Returning to the problem of how this approach to standard is to be 
effected, Henrici's mechanism can be largely considered a search for 
cause. He assumes that, "Behind every variation from standard cost 
there is a reason in operating conditions — and very often an apparently 
good reason. "2 The size and trend of the variances direct supervisory 
attention to certain phases of activity and the causes of an unfavorable 
variance are ascertained. These causes are generally assumed to be 
remediable or nonremediable; "corrective action" is taken in the former 
case, whereas the latter is dismissed or excused in one form or another. 

What has been described above is the essence of so-called "principle 
of exceptions" or "management by exception." It should be noted that 
a step has been taken in this thesis in the development of systematic 
search techniques for the finding and correcting of "remediable causes" 
of higher costs (or alternatively, lost profit) including the determination 
of priorities in the order of search. This work is described in Chapter 5. 
However, the process of "following up" unfavorable variances would 
seem to be only part of the gain which might be achieved from a system 
of budgets or standards. 

The process of investigation per se places the burden of proof upon 
management to discover the cause of variances. This is partially trans- 
ferred to the manager or department head within whose jurisdiction 
the unfavorable variance occurred; e.g., a report of explanation is 
required of him so that he must hunt for causes, or at least reasons, to 
enter in such a report. Often such reports are used to initiate or justify 
a requested change which, if granted or acceded to by "higher" manage- 
ment, will allow the department head to eliminate or reduce the reported 
causes of trouble. Alternatively, the report may focus on the existence 
of some "uncontrollable factor." In principle it may show inefficiency 
as one root of the difficulty, but here psychological (or economically 
rational) factors are likely to enter to cloud or obscure matters so that 



Wide Robnett, Hill, and Beckett (66), p. 431; Lang McFarland and Schiff (46), 
Chapter 16, especially p. 320; and Heckert (34), p. 171 for only a few instances of 
application of this criterion. 

2(35), p. 154. 



12 Preliminary Considerations 

recourse must generally be had to other sources; e.g., bolstering of the 
controls by independent (internal) auditors, special studies, etc. Here 
again variations in report content as well as differing sources of infor- 
mation are utilized to achieve (or to attempt to achieve) ''control." 

Consider once more the problem of control as it depends on the 
motivation (or other psychological and organizational) factors as they 
affect the person who "causes costs to happen" in the first instance. 
The process of consideration may well commence with the setting or 
changing of a standard. The setting of the standard is not sufficient of 
itself to assure or even invite compliance. The problem of directing 
activity toward a goal is one of ''motivation;" a problem which is 
ignored, by and large, in the cost accounting and budgeting, except 
insofar as it deals with issues such as understanding (or the lack thereof) 
of accounting reports by others. However, as the psychologist Ruch 
points out, motivation is an integral part of goal-striving activity:^ 

In any activity there are certain internal conditions or forces 
without which there would be no activity [italics supplied]. These internal 
conditions [motives] serve to direct the organism toward certain goals, 
regardless of whether these goals are, at the time, present in the 
organism. 

This viewpoint (i.e., when there is no motivation there is no activity) 
is fundamental to much of psychology. To bring the matter somewhat 
closer to this thesis, it would appear that Ruch implies that a budget or 
goal, even if externally imposed, must receive some internal recognition 
if it is to be at all effective. The following quotation from H. J. Leavitt 
may be utilized to develop this line of thinking more fully. Professor 
Leavitt notes i^ 

No matter how much power a changer may possess, no matter how 
"superior" he may be, it is the changee who controls the final change 
decision. It is the employee, even the lowest paid one, who ultimately 
decides whether to show up for work or not. 

It may be perceived from the foregoing remarks that a major area 
for investigation of the means of budget control involves the relationship 
between motivations and budgets and standards considered as goals. 
Chapters 2, 3, and 4 contain the results of an investigation into those 
aspects of budget control which can be treated as instances of motivated, 
goal-oriented human behavior. 



1(68), p. 105. 
2(49), p. 132. 



Preliminary Considerations 13 

1.5. Some Remarks on Budget Control and Economics 

This is not the place to discuss at length the relation of business 
practice, budget (and accounting) literature, control, and economic 
theory. However, remarks are made from time to time on these topics 
in various parts of this thesis, so it may be well to summarize at least 
some of the background of these remarks in a more systematic form. 
First, the theory of economics (at least in its standard form) tends to be 
oriented towards the analysis of markets, inter- and intra-industry 
structure, and aggregative types of behavior which are of economy-wide 
concern. For this reason this theory has little to say about the internal 
structure of firms or the problems of control that are observed therein. 

The model of the firm which economic theory supposes is exceedingly 
simple, although perhaps adequate for the purposes for which it has been 
conceived. Briefly, it consists of an ''entrepreneur" who makes the 
basic decisions and certain ''factors of production" (e.g., "labor" and 
"capital") who execute these decisions in faithful detail, and without 
misunderstanding or possible conflict, to whatever degree approximation 
is required. All this is done at a price which if paid by the entrepreneur, 
produces these factors (fully instructed) in whatever quantity the 
entrepreneur requires. Uncertainty, if introduced into the anlaysis, is 
never allowed to intervene within the firm, but it is allowed (in many 
cases) to be a part of the problem with which the entrepreneur must deal 
as he assesses his external (economics) environment: markets, GNP, 
and possible actions (or reactions) of his competitors. 

Driven usually by an assumed goal of maximum profit (long- or 
short-run), the entrepreneur chooses a "production function" — a relation 
which determines the amounts of various factor inputs required to 
produce the levels of outputs which are deemed desirable by the entre- 
preneur. This function, in any given state of technological knowledge, 
is assumed to be chosen optimally. Various other details then follow, 
as summarized in the following statement from P. A. Samuelson:^ 

I. The first fundamental assumption is that the firm tries to maximize 
its profits, and from this the following internal conditions of equilibrium 
can be deduced. 

A. Any output which is produced must be produced with factor 
combinations such that total cost is a minimum. As a result of this 
we have two corollaries. 

1. The marginal productivity of the last dollar must be equal in 
every use. 



K69), p. 88. 



14 Preliminary Considerations 

2. The price of each factor of production must be proportional to 

marginal physical productivity, the factor of proportionality being 

marginal cost. 

B. That output will be selected which maximizes net revenue, total 

cost being optimally determined by the previous conditions. This 

implies 

1. The equality of marginal cost and marginal revenue, the slope 
of the latter being the smaller. 

2. In combination with previous conditions under A we also have 
the marginal value productivity of each factor equal to its price, 
the first term being revenue times marginal physical productivity. 

3. Total cost must not exceed total revenue, since otherwise the 
firm would go out of business. 

Evidently this theory is then concerned with certain kinds of machin- 
ery for calculating costs as a prelude to effecting decisions for planning 
resource allocations and, except for the assumption of faithful and 
compliant factors, it has little to do in a direct way with the topics of 
control which are primarily at issue in this thesis. Other points may also 
be made which deal with dynamic as well as static aspects of this theory. 
However, these are points of increasing refinement, whose pursuit would 
lead the discussion too far astray. One other general point should be 
made, however. Under the doctrine of perfect competition there is a 
control (or at least some aspects of a control) which can be highly 
effective under certain circumstances. The two important aspects (or 
rather assumptions of this theory) are as follows: (1) freedom of entry 
and exit into an industry (anyone may become, or cease to become an 
entrepreneur), and (2) no firm is sufficiently large so that its actions can 
have any influence on the prices it pays or receives. 

The last of these two assumptions is designed to supply an ''imper- 
sonal" force without regard (it might be said)^ to the motivation of the 
individuals subject to this force. The first admits, under certain plausible 
conditions, of a series of new entrants who will be attracted into this 
industry whenever conditions warrant (e.g., because some entrepreneurs 
are not wholly efficient) and allows not only for new^ competition from 
this source but also for competition from already established entrants 
who may expand their activities at the expense, possibly, of others.- 



^It must be assumed that there is always a "sufficient" supply of persons willing 
to become (or try to become) profit maximizing entrepreneurs. Note, however, that 
it is not necessary for all entrepreneurs (or potential entrepreneurs) to have this desire. 
A "marginal" group will, in principle, suffice. 

2Cf. F. H. Knight (43), pp. 282 ff. for one of the few places where the question of 
"supply of entrepreneurship" is dealt with. 



Preliminary Considerations 15 

This means, when translated into administrative terms, that under 
suitable assumptions each such market enforces (at the top)^ a rapid 
series of promotions, demotions, firings, hirings, etc., which along with 
other penalties and rewards are proportioned to degrees of accomplish- 
ment and failure. This is all done in a highly effective and, moreover, 
impersonal manner whenever these assumptions are fulfilled. 

In admiration of this system — or perhaps for other reasons — 
certain business firms have tried (with more or less success) to import 
various facsimiles of this system as integral parts of their administrative 
(and control) mechanisms. The so-called internal profit-and-loss control 
systems are cases in point. ^ Among other things departments are 
allowed to deal with each other and with outside entities more or less 
directly,^ submitting and receiving bids, negotiating transfer prices, etc., 
each in the pursuit of its own optimum. 

Such systems have not always been successful. On the one hand, 
it is not always true that the increased efficiency of each department will 
add up to an increased efficiency of the whole entity (e.g., as measured 
by profit). On another hand, the assumptions of the theory of perfectly 
competitive markets have not always been attended to in any careful 
manner. Moreover, it is by no means clear that they are capable of 
realization in any place but an ''external" market. For instance, there 
are products within a firm (e.g., those dealing with "work-in-process") 
which have no readily available sources of supply (or demand) other 
than different departments within the same firm. Such departments 
are subject to exploitation by these other departments, at worst, and 
are insulated from outside pressures, at best. Moreover, departments 
vary in size, and some may be of such preponderant importance as to 
vitiate one or both of the key assumptions in a serious way. This again 
may lead to exploitation or inefficiency, factors which are further re- 
enforced by the appearance of a multiple hierarchy within the firm 
whose existence has no place in a perfectly competitive market as 
conceived in economic theory. Finally, the fluidity of entrance and exit 
which "allows anyone to become an entrepreneur (at the expense, 
perhaps, of everyone else)" as his own willingness and judgment dictate 
does not adjust easily to the systems of promotion, demotion, and 
dismissal found in the hierarchies of most manufacturing firms that are 
of reasonable size. 



^I.e., at the head of each enterprise. 

2Cf., for example, Peter Drucker (28) for a description of some of the General 
Motors Corporation's endeavors in this direction. 
^E.g., via a central purchasing or sales office. 



16 Preliminary Considerations 

In any event this economic market theory is not germane, for the 
most part, to this thesis, except for possible quahfications and points of 
relevance that will be introduced at appropriate points in the text. 
Pending their introduction it may be said that the main topic of the 
thesis is concerned with the effects of goals held by individuals (these will 
be called aspirations), goals held out for them (or imposed upon them 
by others), various kinds of reports, and the results these may have for 
actual performance — immediately and ultimately — under various 
assumptions. 

The presentation will be as follows. In Chapter 2 a mathematical 
model of a one-cost, one-department budget situation is developed in 
which the behavior of the individual budgeted is assumed to be a 
composite of several psychological propositions. Chapter 3 contains 
descriptions of several experiments with their underlying theories, which 
are to some extent germane to the study of budget control, and a very 
limited discussion of a practical application. Chapter 4 contains the 
description and results of an experiment designed to test some of the 
assumptions which are made in current budgetary practice and also the 
assumptions about human behavior upon which the mathematical model 
of Chapter 2 is based. Chapter 4 also relates the results of the experi- 
ment, wherever possible, to the findings of other psychological studies. 
In Chapter 5, the assumption of a one-cost model is dropped and a model, 
designed primarily for the purpose of studying the processes of co- 
ordination and planning (although having additional implications for 
control), is presented. 



CHAPTER 2 



A Mathematical Model of a 
Budget Control System 



2,1. Introduction 

Having established the need for a particular kind of budget whose 
aim is control, as opposed to planning or forecasting, it is now desirable 
to investigate the relationship between the control budget figure and 
actual performance. 

To re-emphasize a point made in Chapter 1, a "good" control budget 
is one which produces "good" results. If it is desired to minimize cost 
in a given department, and if a budget of $1000 produces a cost of $1001, 
and a budget of $300 produces a cost of $1000, the latter is a better 
budget. The magnitude of the budget figure is unimportant other than 
in terms of its impact on cost. 

The budget is a goal imposed on an individual, who shall be called a 
"department head," by his supervisor or supervisors (management). 
To its attainment are occasionally attached positive rewards, but more 
frequently, negative rewards are attached to its nonattainment. If it 
could be assumed that the department head took the budget as his 
personal goal and worked toward this goal with maximum effort, the 
criterion of budget control for a single individual would be trivial — i.e., 
choose a cost goal at the technological minimum for the operation and 
let him work toward it. It is not difficult to visualize the effects of such 
a goal in practice. If there is negative reward attached to its non- 
attainment, some change must be made in the system or the department 
head will resign, be discouraged, or possibly simply sabotage and oppose 

17 



18 A Mathematical Model of a Budget Control System 

the system, perhaps soliciting the help of others to form a group for this 
purpose.^ Regardless of the amount of positive reward attached to its 
attainment, the expected value of reward, statistically speaking, is zero; 
and the net expected value of rewards and penalties is negative. 

In practice the budget may exist on paper at the technological 
minimum, and doubtless some budget or engineering departments may 
make just such forecasts for their own guidance. But in actual execution 
it is usual to secure assent of persons who are to be controlled so that 
some deviation or adjustment may be applied to this figure. This means 
that there is some ''acceptable" level of cost which, in general, will be 
above the theoretical optimum. If cost descends below this level the 
performance is rated as meritorious, but if cost is above this level then 
there is an implied criticism which may receive explicit form when this 
fact is called to the department head's attention, and an investigation 
of causes supplemented by a report (perhaps by outsiders) may ensue. 
Reprimands, promotion passover, and dismissal are possibihties. 

It is a postulate of this thesis that unwritten ''acceptable levels" are 
the common bases of control budgets. Alternatively, there may be an 
acceptable rate of approach to the technological optimum, which in 
practice becomes the control element, and the cost level is then deter- 
mined from the acceptable rate of improvement of the control budget. 

Another hypothesis whose logic and empirical content will be inves- 
tigated is that a stationary budget"^ is not an effective control budget. If the 
budget level is never attained, then some other criterion is in fact 
replacing it as a control element. If the level is consistently attained, 
the question of the possibility of consistently obtaining operation at a 
lower cost will never be answered, because there is no incentive to 
improve performance. If a level is obtained part of the time, either it 
must drift toward consistent attainment or nonattainment, or the 
percentage of the time it will be attained will become stable at some 
value which produces an acceptable balance of positive and negative 
reward for the department head. Another and related issue is whether 
anyone whose performance displays such characteristics would strive 
for the same reward balance at a lower budgeted cost level. It is part 
of the task of this thesis to provide a formal basis against which such 
questions may at least be asked (as they are not in the present literature) 
and thereby provide a start towards a formal theory of budgetary 
control. In particular, it is proposed to deal minimally with the question 



iCf. Argyris (2). 

21. e., a budgeted cost for a given operation which does not change over time. 



A Mathematical Model of a Budget Control System 19 

of budgetary level setting in order to focus on the dynamics which center 
about the question of when (and how) a budget should be changed. 

2.2. The Budget as a Determining Factor in Formation of Aspi- 
ration Levels^ 

When management presents the department head with a budget, it 
can only present its goal. It is a hypothesis of this thesis that manage- 
ment can increase the tendency of the department head to aim at or 
below this goal by increasing the positive reward associated with its 
attainment and/or increasing the negative reward associated with its 
nonattainment. 

Management can enforce absolute compliance with the budget by 
dismissal for noncompliance. After this policy has been in effect for a 
short time, management would retain only the department heads who 
aimed at or below the budget and were successful at achieving their aims. 
It would seem, however, that for this procedure to be in operation 
without a decimation of supervisory personnel, the budget levels would 
need to be set far above expected cost in order to allow for random 
fluctuations. Such a procedure seems unlikely to cause the department 
head to drive his costs far below the budget, since safety will take priority 
over innovation. The fear of a lowering of the budget if he performs 
too well will undoubtedly dominate a desire to impress management with 
superior performance. ^ Barring this undesirable procedure, the budget 
may be considered at best a candidate for the department head's goal 
but more generally as one factor which operates in its determination. 
This level of cost toward which the department head strives will be 
termed his aspiration level. 

2.3. Existing Models of Aspiration Level Determination 

The definition of aspiration level which will be used here is consistent 
with the definition of J. D. Frank: ''The level of future performance in a 
familiar task which an individual, knowing his level of past performance 
in that task, exphcitly undertakes to reach. "^ Considerable documen- 
tation, both theoretical and empirical, is available for the existence of 



^My initial contact with aspiration levels was aided immensely by William H. 
Starbuck and his excellent survey of the field (74). The mathematical formulations 
of this section (before budget impositions) are contained in that paper. 

2Cf. Barnard (3), particularly pp. 149-153. 

3J. D. Frank (33), p. 119. 



20 A Mathematical Model of a Budget Control System 

aspiration levels in the absence of explicit external goals. For example, 
Lewin, Dembo, Festinger, and Sears ^ have hypothesized a model in 
which they define as the basic reference variable, x, the level of difficulty. 
The probability of success, P(x), is the subjective probability of achiev- 
ing a level of performance of equal or greater difficulty than x. The 
utility of success, S{x), and the utility of failure, F{x), are assumed to be 
monotone increasing and decreasing functions of x, respectively. The 
level of aspiration, A, is that level of difficulty which maximizes V{x)y 
where 

(2.3.1) V{x) = P(x) • S{x) + [1 - P{x)] • F{x) 

If X is a monotone decreasing function of c, the cost level, then this 
utility maximization could conceivably be used to find the aspired cost 
level. In the industrial context, however, it is difficult — almost impos- 
sible — to envision a situation in which a department head does not 
perceive some sort of external goal whether or not explicitly stated. 
Imposing upon the Lewin et al. model the cost transformation and a 
budget level, h, for which reward is administered on a go no-go basis 
produces the following discontinuous utility function : 

(2.3.2) V{c) = Pic) ■ S(c) -h [1 - Pic)] ■ Fie) +R c<h 

= Pic)-Sic)+[l- Pic)]-Fic)-\-P Oh 

where R is the utility of reward for attaining the budget, P the disutility 
of the punishment for not attaining the budget, and Pic) is the proba- 
bility of a cost level less than or equal to c. The point of maximum 
expected utility (the aspiration level) of the discontinuous function will 
occur at the same point. A, as that of the continuous function (i.e., 
R = P = 0)iih>A (in which case the budget is superfluous), or if 
h < A and i^ — P is less than the difference between ViA) and T^(6) 
calculated for i^ = P = (which generally speaking will result if the 
subjective probability of success at the budgeted level is small relative 
to the reward for attaining it). Otherwise the aspiration and budget 
levels will coincide. ^ 



iSee (37b) pp. 356-377 but especially pp. 360-361; and also for further discussion 
of the model, Starbuck (74), pp. 1-3. 

2The aspiration level need not be single valued. If ^(^)22p = o ^ ^^^^^r,p,=q 
b < A, two values appear: 

a = b 
a = A 

Although, mathematically, this case occurs with zero probability, the range of 
indifference between b and A as aspirations is undoubtedly finite and perhaps large 
in practice. 



A Mathematical Model of a Budget Control System 21 

An obvious implication of the augmented Lewin model is that 
management can equate aspiration and budget levels for sufficiently 
large but finite R — P, at any value of the budget including (unfor- 
tunately) 6 = (unless F{0) = — co , which would render the entire 
model meaningless). The model also implies that either the aspiration 
level is equal to the budget or is completely independent of it. This 
conclusion stems from the go no-go character assumed for reward; the 
more reasonable belief that an individual's aspiration is affected by the 
budget even if the two are unequal stems from the implicit assumption 
that the reward-penalty structure produces discontinuities in the utility 
function at more than one point and that the locations of the additional 
points depend upon the budget. Proliferation of these points of dis- 
continuity would provide an aspiration level at one of them or at A. 
With the exception of the allowance of an aspiration level of zero cost, 
the augmented Lewin model predicts plausible behavior of the aspiration 
level in response to a budget. 

The difficulties of the Lewin model for this exposition are twofold. 
First, it falls short of supplying all of the required characteristics for 
operational significance in that it provides no means of predicting actual 
behavior. Secondly, the rehance on subjective probabihties makes it 
more difficult operationally to predict the aspiration level than to 
measure it. These difficulties preclude the adoption of the Lewin model 
for the purposes of this chapter. The implications of the model to be 
presented below are, however, consistent with those of the Lewin model. 

Two models, one of Simon ^ and one of March and Simon, ^ utihze 
the availability of alternatives as a factor in the determination of 
aspiration levels. 

In the Simon model, phrased in the terminology of the augmented 
Lewin model, 

(2.3.3) 7(c) < for c > a 

y(c) > for c < a 

where a is the aspiration level in terms of the aspired cost level. Essen- 
tially, Simon assumes that individuals do not maximize utility, but 
merely search until a satisfactory goal (one with positive utility) is 
found, and then cease to search. The aspiration level then is the last 
goal found by the search process; all earlier goals encountered are 



iSee Models of Man (72), pp. 241-260. The mathematical formulation of the model 
in Lewin terminology is due to Starbuck (74). 
2March-Simon (57), p. 48. 



22 A Mathematical Model of a Budget Control System 

unsatisfactory (possess negative utility). According to this hypothesis, 
if the budget (which is an alternative which requires no search) possesses 
a positive utility, the department head will adopt it as his aspiration 
level. Using the expected utility function of the augmented Lewin model 
(2.3.2), it is noted that the Simon model equates the aspiration and 
budget levels in every case that the Lewin model does unless V(c) < 
for all c (in which case the Simon model has no solution for the aspiration 
level while the Lewin model has a solution which may equate a and b) , 
and also some in which the Lewin model would choose a = A (i.e., 
V{A) > V{b) > 0). The Simon model, like the Lewin model, gives no 
clue as to the reaction of performance to changes in aspirations. Further- 
more, since it does not specify the method of search if V(h) < 0, it leads 
to the completely plausible but operationally limited conclusion that 
either aspiration level equals the budget level or is unknown. 

The March-Simon model is, to my knowledge, the only existing 
dynamic model of aspiration level determination. It can be summarized 
in the four equations:^ 

(a) ^ = a{R- A+a), a > 0, a > 
at 

(b) S = R - A 

^^•^•^^ (c) L = ^(S- S) ^ > 0, /3 > 

(d) ^ = a{L-h-cR) a>0,h>0,c>0 

at 

where S = satisfaction, A = level of aspiration, L = search rate, R = 
expected value of reward, and S = desired level of satisfaction. The 
system possesses a stable equilibrium solution: 

(a) A = R +a 

^^•^•^^ (b) L = h+cR 

It is to be noted that the aspiration level can be interpreted as aspired 
reward, as opposed to the other models in which aspiration level is related 
to a performance level or the difficulty of its achievement. It is much 
more reasonable to assume that the long-run aspirations of an individual 
whose behavior these various models attempt to describe relate to his 
ultimate compensation rather than his performance. However, in the 
absence of some way of relating the level of search effort to a measurable 
set of criteria, the above model is of limited usefulness as a basis for 
the principles of a theory of budgetary control. 

Although the above discussion is not by any means an exhaustive 

Ubid. 



A Mathematical Model of a Budget Control System 23 

coverage of aspiration level models, I have described those which I feel 
have the greatest applicability to aspiration level formulation with the 
compHcation of an external goal. Each, however, lacks at least one 
essential attribute for the present task of relating aspiration levels, 
external goals, and actual performance. I therefore proceed to the 
development of a new model which occasionally borrows from the ones 
described in this section and, wherever possible, is consistent with them. 



2.4. Scope and Postulates of the Model 

The aim of this model is to provide a vehicle for establishing rules 
for optimum management policy with regard to the setting of budgets 
for a single department. The choice of a one-department model requires 
for its validity an amendment to the definition of Chapter 1 which 
would allow a ''sub-budget" of the budget control system for the enter- 
prise to be considered by itself. This dissection leaves out many interest- 
ing and significant questions concerned with interdepartmental reactions 
and therefore some of the raison d'etre for the existence of a central 
budgetary office. Some of the facets of the interrelationships will be 
treated in Chapter 5, but are omitted here for the purpose of investi- 
gation of the unicellular case whose behavior is as yet incompletely 
determinate. 

It is assumed that the accounting system and the budget are so 
devised that the behavior of cost in a single cost-incurring unit may be 
considered ceteris paribus. That is, a department head can be held fully 
responsible for the costs incurred; overhead costs for which he cannot 
be held accountable, variations in production level which are not of his 
making, etc. have been removed from the calculation of cost. There is, 
incidentally, considerable indication that the accounting procedures for 
measurement of costs and the allocation of responsibility for them are 
highly developed,^ and that a single ceteris paribus cost figure can be 
attributed to the operations of a particular man in the organization who 
will generally accept responsibility for its fluctuations. The problem 
which is at issue here is not in ascertaining what costs are, but what goals 
are to be set for them and how the cost performance should be evaluated. 
It is further assumed that the costs discussed here either (1) occur under 
invariant output or (2) have been adjusted independently for variations 
in output. 

Three interrelated cost levels are relevant to the system. The 



^See, for example, Lasser (48), p. 34. 



24 A Mathematical Model of a Budget Control System 

budgeted level of expenditure is that level which is set by higher manage- 
ment. The cost level which the department head strives to achieve — 
the aspired level — is dependent upon the budgeted level, but is not 
necessarily equal to that level. The level of expenditure which the 
accounting system shows to have actually taken place (perhaps refined, 
as indicated in the preceding paragraph) will be known simply as the 
expected actual cost level, or more simply, the cost level. 

The level of expenditure is in units of dollars per fixed accounting 
period; e.g., dollars per year. However, the level of expenditure can be 
determined at any instant of time; i.e., it is the instantaneous rate of 
flow of expenditures. 

The assumptions of the model can best be comprehended in the 
following set of postulates for the behavior of a hypothesized depart- 
ment head. 

(i) If there is a discrepancy between the expected actual level of 
expenditure and the aspired level of expenditure, he will attempt 
to reduce this discrepancy by moving his aspiration level toward 
the actual level at a rate which depends on the size of the dis- 
crepancy. 

(ii) In addition to the effect caused by the discrepancy, the aspired 
level of expenditure will be lowered in response to a lowering of 
the budgeted level of expenditure. 

(iiia) The department head will be encouraged if the discrepancy (actual 
expected cost minus aspired cost) does not exceed some positive 
value known as the discouragement point. 

(iiib) The department head will be discouraged if the discrepancy 
exceeds the discouragement point but does not exceed a larger 
value known as the failure point. 

(iiic) If the value of the discrepancy exceeds the failure point, the 
system will cease to exist, or a new one will come into being; 
''the department head will resign." 

(iva) If the department head is encouraged, he will attempt to reduce 
a positive discrepancy by reducing the expected actual level of 
expenditure; he will react to a negative discrepancy by allowing 
expected cost to rise. ^ The rate of reduction or increase depends 
upon the size of the discrepancy. 

^If a multiple commodity or multiple cost structure was hypothesized, it would be 
assumed that negative stress in one area would direct attention to another. Cf. 
Edwards (30), and note Chapter 3, p. 54, of this thesis. 



A Mathematical Model of a Budget Control System 25 

(ivb) If the department head is slightly discouraged, he will reduce the 
discrepancy by reducing expected cost at a lower rate relative to 
a given discrepancy than he would if encouraged. If he is 
moderately discouraged, he will allow expected cost to increase, 
but at a sufficiently small rate that the discrepancy will not be 
increased. If he is extremely discouraged, he will allow expected 
cost to increase at a rate which increases the discrepancy. 

Postulate (ii) describes a situation in which the budget is a figure 
about which there are several auxiliary points, each of which defines a 
particular set of rewards. Using the Simon model of aspiration level 
determination, the department head will find a point at which the 
rewards are ''satisfactory." He will then study the relationship of this 
point to the budget, find out about how much it changes for a given 
change in the budget, and then change his aspiration level accordingly, 
responding to changes in the budget. Postulate (i) is essentially Lewinian 
in nature, in that the department head may be interpreted as responding 
to a positive discrepancy as a reduction of the perceived probability of 
success of the original aspiration, adjusting his aspiration level in the 
direction of increased probability of success. The discrepancy between 
the expected actual level of expenditure and the aspired level of expend- 
iture is appropriately termed a measure of stress, since clearly the 
department head's "emotional tension, produced by frustration, "^ varies 
with the size of the discrepancy. A compromising of goals is a well- 
known reaction to stress, ^ and hence postulate (i) may be interpreted 
directly as a stress-reducing mechanism without considering the existence 
of subjective probabilities. 

Postulate (iva) describes the department head as exhibiting another 
"normal" form of reaction to stress — striving to improve performance. 
A primary assumption of this model is that man is an improvable animal 
and that, given sufficient motivation, cost reduction is a possibility. 
March and Simon (57) have confined their discussion of improvement to 
an increase in search behavior.^ It is assumed here that improvement 
is possible through increased experience with the task, diverting of 
effort from nonorganizational goals, development of increased "cost- 
consciousness" (diverting of effort from other organizational goals in 



IF. S. Ruch (68), p. 154. 

"^Ibid., p. 162. 

^But Cf. Handbook of Experimental Psychology, Chapter 13, Neal E. Miller, 
"Learnable Drives and Rewards," where this same kind of search for improved 
situations is ascribable to fear and anxiety (78a), 



26 A Mathematical Model of a Budget Control System 

which there is less stress), or mere harder work — all of which may be 
considered part of or in addition to search behavior. Postulate (iiia) 
notes a limitation on the amount of stress which the department head 
can tolerate and still devote his efforts to cost reduction at maximum 
effectiveness. 

Postulate (ivb) describes the various stages of withdrawal within 
the range of stress denoted by postulate (iiib). Caused by sublimation 
or ineffective effort due to stereotypy of response, the department head 
will be less successful in reducing costs. The neurotic response of extreme 
discouragement will eventually lead to ultimate withdrawal (postulate 
(iiic)), provided some change does not occur within the system. The 
assumption that exceeding the discouragement point will evoke one of 
only three types of behavior depending on the individual department 
head is made for the sake of simplicity rather than necessity. 

2.5. Mathematical Formulation of the Model 

An analytic statement will help to clarify what is involved in the 
preceding postulates, and for this purpose it will be assumed that the 
relationships between the variables are linear. Let c = expected actual 
level of expenditure, a = aspired level of expenditure and h = budgeted 
level of expenditure. Then 

(2.5.1) ^ = ^(c - a) + 7 ^, where /3 > 0, 7 > 

at at 

describes the relationships of postulates (i) and (ii), and 

dc 

(2.5.2) — = —aAc — a) where z = 1, 2 and ai > a2, ai > 
at 

describes the relationships of postulates (iva) and (ivb) with the excep- 
tion of limitations on a 2. 

Let M = the discouragement point and A^ = the failure point. The 
constants of (2.5.2) can then be qualified as 

(2.5.3a) ai = ai where c — a < M (postulate iiia) 

(2.5.3b) ai = a2 where M < c — a < N (postulate iiib) and 

(2.5.3c) ai not defined where N < c — a (postulate iiic) 

for M, N > and finite. The state of discouragement depends upon ^ 
as well as a 2 in the following manner; 



A Mathematical Model of a Budget Control System 27 

(2.5.4a) if /3 + 0:2 > 0:2 > he is slightly discouraged 

(2.5.4b) if /3 + a2 > > 0:2 he is moderately discouraged and 

(2.5.4c) if > ^ -\- a2 > a2 he is extremely discouraged 

in accordance with postulate (ivb). 

This completes the mathematical formulation of the system. 

An assumption which will be made about management budget 
behavior is 

(2.5.5) ^ = -Kj (j = 1, ..., n) 

which postulates a constant rate of budget reduction which can be 
changed to another rate by a management decision. That is j = 1, 2, •••, 
n represents the set of decision possibilities with respect to rates of budget 
reduction. Note that db/dt is (at least in part) controlled by persons 
beyond the authority of the affected manager. 

Provided either of the conditions (2.5.3a) or (2.5.3b) are met and 
no management decisions occur for all t, denoting the initial values of 
the variables by a zero subscript, the solutions to the system of equations 
(2.5.1) and (2.5.2) are 

(2.5.6a) c = Coe-(^+«i>' + ""'l' ^ ^^' [1 - e-^^+-i^'] 

^^■^^^- [e-(P+ai)t + (/3 + adt - 1] 



(0 + «,) 



The system possesses a stable equilibrium if and only if jKj = 0, 
which will result either if the budget is static or if the department head 
completely ignores its changes. In this case, the equilibrium solution is 

(2.5.7) lim c = lim a = — -— 

provided, of course, that no changes in the department head's state of 
encouragement or management decisions take place. 

However, a basis for defining the behavior of the system as t grows 
large exists since 

(2.5.8) I = - a,(c - a) 

= - a,(co - a,)e-^^+<'0' - ^^^ [1 - e-^^+^i^'] 



28 A Mathematical Model of a Budget Control System 

possesses a limit provided that (/? + ai) > 0. Specifically 
(2.5.9) lim ^' - ^'^^^ 



dt iS + a, 



Note, then, from the last expression that external (e.g., central) manage- 
ment can produce an effect on cost behavior even when Oo = Cq. Also, 
the expression (2.5.9) suggests the usefulness of an equilibrium rate for 
the case dc/dt and da/dt constant. In terms of such an equilibrium 
{13 + av) niay be regarded as the "speed of adjustment" or "time 
constant;" i.e., the rate at which dc/dt approaches rate equilibrium. 
(A constant dc/dt implies constant da/dt and conversely, so that it will 
only be necessary to discuss dc/dt.) 



2.6. Behavior of the System in Response to a Specific Rate of 
Budget Reduction 

A system which can attain a condition of rate equilibrium in a state 
(of encouragement or discouragement) will be considered rate stable in 
that state. A system in a given state which is both rate stable in that 
state and for which 

(2.6.1) lim ^ < 

f^co at 

will be designated an admissible system. By definition, an admissible 
system is a sufficient condition for nonincreasing costs in the limit. In 
order to show that an admissible system is also a necessary condition, 
it must be shown that dc/dt is bounded beloAv.^ If jS + a,- > a, > 0, 
as in the encouraged state and the slightly discouraged state, the 
relations «» > and {c — a) < N, ensure that^ 

(2.6.2) Tt^ ~ ""'^^ - «) ^ - «»^ 

On the other hand, if /3 + a2 > > 0:2, as in the moderately discouraged 
state, then 

(2.6.3) 5^ = a2(a - c) > a2{-N) > 0. 
at 



^Otherwise, the possibility would exist that a system which possesses no rate 
equilibrium solution could exhibit cost reduction at an ever-increasing rate. 

2It is interesting to note that should the pathological case c — a < exist, then 
dc/dt is everywhere non-negative. If ai < were allowed, however, this non- 
negativity would not hold and, in fact, dc/dt would not be bounded below. 



A Mathematical Model of a Budget Control System 29 

The extremely discouraged state is likewise devoid of admissible systems 
since 

(2.6.4) ^ = a^ia - c) = {-a2){c - a) > (-^2)^ > 0. 

It follows immediately that if a system remains in a state but does 
not possess rate stability in that state, dc/dt, since it must then be 
unlimited but bounded below, will increase without bound. Thus an 
admissible system is both a necessary and sufficient condition for 
nonincreasing costs in the limit. 

If the department head is initially encouraged, which means that 
the initial spread between actual and aspired cost behavior is sufficiently 
small, the system will come to rest in the encouraged state if management 
does not attempt to reduce costs at too great a rate. That is, if manage- 
ment chooses to keep the department head encouraged it will choose a 
rate of budget reduction Kj such that 

(2.6.5) < yKj < (/? + ai) M 
Since /? + a: 1 has been assumed non-negative 



V^.u.ua; 


U, — \CQ UQJt 


' ' (^ + «i) 


(2.6.6b) 


< (co - ao - 


- M)[e-(^+«i)'] -\- M 


(2.6.6c) 


< M 





Not only is management thus assured of keeping the department head 
encouraged, but it is also assured, in the long run, of a constant rate 
of cost reduction since 

(2.6.7) li^ I = _ ^ < 0, 

which clearly describes an admissible system. If the first weak in- 
equality in (2.6.5) is replaced by the stronger condition of strict in- 
equality then a nontrivial admissible solution is assured.^ 

If, however, management elects a high rate of cost reduction, seen 
by the department head to be "intolerable," viz. 

(2.6.8) yKj > i(3 + ai) M 



iThis obviously requires positive 7 as well; i.e., some degree of acquiescence from 
the department head. Cf., pp. 9-12. 



30 A Mathematical Model of a Budget Control System 

the solution obtained for t by equating equation (2.6.6a) to M, 

yKj- (/3 + aO(co 



(2.6.9) ^, =:^_ln 

P -\- Oil 



yKj- (/3 + ai)M 



A] 



is non-negative, since 

(2.6.10) yKj - (^ + ai){co - ao) > yKj - (^ + a,) M>0 

The trivial solution, n = 0, (immediate transition to discouragement) 
can occur only if the initial conditions place Co — ao precisely at the 
discouragement point M. A non-negative solution for (2.6.9) implies 
that the system will undergo a transition to the discouraged state 
immediately or after a finite time, ri. 

If the department head is initially discouraged {N > (cq — Qq) > M), 
the conditions for rest and transition are dependent on the level of 
discouragement. If the department head becomes extremely discour- 
aged, and if management unwisely applies a rate Kj such that 

(2.6.11) yKj> (/3 + a2)(co-ao) 

then it can defeat its own purpose and cause a cost increase. This occurs 
because 

(2.6.12) c-a= {co- ao)e-(^+«.)' + -r^^ [1 - e-^^^-.'^] 

p -1r ct2 






yKj 



a2 



is an increasing function of time. Furthermore, equating equation 

(2.6.12) to A^ gives a non-negative solution for t, analogous to the 
discussion of (2.6.9) above, except that the relevant summary of con- 
ditions is 

(2.6.13) yKj - (/? + a2)N > yKj - W -\- «2)(co - flo) > > ^ + a. 

Hence, the management behavior shown in (2.6.11) assures the eventual 
failure of the system — i.e., "the department head will leave the organ- 
ization" — and does so under the adverse condition (for management) 
of causing an increase in costs to bring this eventualit}^ about. If 
management wishes to avoid the transition from extreme discouragement 
to failure, it must choose a rate of budget increase which is sufficienth^ 
large to overcome the disequilibrium-producing behavior of the depart- 
ment head and the initial discrepancy; namely, 

(2.6.14) yKj < (13 + a2)(co - ao) < (/3 + a2)M < 



A Mathematical Model of a Budget Control System 3 1 

The second and third strong inequahties are required by the negativity 
and the initial conditions. In equation (2.6.14) imphes 

(2.6.15) iS + a2 < < (/? + «2)(co - ao) - yKj < (/? + as) - yKj 
which insures a positive solution f or ^ to c — a = M which is 

(2.6.16) r. = j-^^ In [ (^ + a.) M -yK, J 

which in turn implies a transition to the encouraged state. 

In summary, under the indicated conditions, management can adopt 
either one of two alternatives which are better than the one indicated. 
It can force such an extremely discouraged department head to resign 
and thereby at least avoid the cost increases which would otherwise 
produce this eventuality anyway; or also it can undertake a restoration 
of the department head's confidence by granting him a large budget 
increase. No intermediate solution exists. 

It must not be assumed, however, that management can either act 
arbitrarily or ignore the cost consequences attendant on budget changes 
for the slightly or moderately discouraged department head. Manage- 
ment can still cause system failure by choosing a rate such that 

(2.6.17) yKj > (^ + a2)N 

The proof of transition to failure is precisely the same as for the transition 
to discouragement above with A^ and a 2 substituted for M and ai, 
respectively. The parameter values of a and /? are, of course, vitally 
important. By making the same substitutions, it can be shown that 
failure is impossible, given 

(2.6.18) (i3 + «2) A^ > yKj > (/3 + az) M 
The second half of the inequality implies 

yKj 



(2.6.19) 



I /? + a2] jS + a: 

>\m - ^^^' 1e-(^+".)^ + M 



> M 



32 A Mathematical Model of a Budget Control System 

Thus the system will come to rest in the discouraged state ^^ith 

(2.6.20) li'" I = - ^ 

which will be an admissible solution if the department head is only 
slightly discouraged, inadmissible otherwise. Without further laboring 
the point, equation (2.6.16) is a positive solution for t of the equation 
c — a = M, assuring transition to the encouraged state provided that 

(2.6.21) yKj < (/3 + az) M 
which becomes 

(2.6.22) (^ + as) (Co - a,) - yK > {(3 -\- 0:2) M - yK > 

With the moderately discouraged department head no admissible 
solution is available so that the permissive procedure sho^^Ti in (2.6.21) 
is advantageous. The slightly discouraged department head, however, 
may perform better in the discouraged state if his failure point is suffi- 
ciently high. 

Specifically, if in the slightly discouraged case, A^ > -^-—, Kj can be 
chosen such that 

(2.6.23) {(3 + a2)N > yKj >-(/? + a2)M 

OL2 

Clearly this condition and ai > 0:2 > insures fulfillment of inequation 
(2.6.18), the condition for rest in the discouraged state, ^ with an admis- 
sible solution given by 

(2.6.24) lim I = - ^ < - «.M 

t->oo at /3 + a2 

In the encouraged state the condition for rest, inequation (2.6.5), implies 
that the rate-equilibrium value of cost reduction has a lower bound; 
namely 

(2.6.25) lim ^^ = - ^ > - aM 

f-»oo at p -\- ai 



It is also relevant that 0:2 ^ implies 



-(3 + „,)_(^ + „0=^i^^^^>0 

a2 «2 

SO that a K which satisfies (2.6.23) also satisfies (2.6.8), the condition for transition 
to discouragement, assuring the attainment of the best solution shown in (2.6.24) 
independent of initial conditions. 



A Mathematical Model of a Budget Control System 



33 



indicating that a higher rate of cost reduction is possible in the discour- 
aged state. This poHcy might be considered ''hard-boiled" or even 
unscrupulous, but the rate of budget reduction for the man who becomes 
only slightly discouraged and gives up completely only under extreme 
provocation should be calculated to keep him just short of resignation 
at all times. 

To bring some of the points already discussed together in a conven- 
ient form, the results of the various management actions are summarized 
in Table 2.1. 

TABLE 2.1 



Initial State 


Management Action 


System Reaction* 




yKj>{0+ai)M 


-^ discouragement 


encouraged 


< jKj < (/3 +aOM 


admissible rest (best 
solution if A'' < — M 
or 0:2 < 0) 




yK, > (iS + ao)N 


-^ failure 


slightly- 
discouraged 


(/3 + a,)N > yKj > ((3+ a,)M 


admissible rest (best 
solution if A^ > —M) 

a2 




yKj < (^ + «2)M 


— > encouraged 




yKj > + a2)N 


— ^ failure 


moderately 
discouraged 


(/3 +«2)iV > yK, > {(3+a,)M 


rest — not admissible 


yKj < (/3 + a,)M 


-^ encouraged 


extremely 


yKj > (/3 + 02) (Co - ao) 


-^ failure 


discouraged 


yKj < (/3 + a 2) (Co - ao) < {fi -^ a2)M 


— > encouraged 



*The symbol -^ represents "transition to." 

If management chooses to restrict itself to only one level of cost 
reduction, the system must come to rest in its initial state, come to rest 
after a single transition, or fail. 

The condition for the encouraged — > discouraged transition given 
by (2.6.8) implies 

(2.6.26) yKj > {(3 + ai)M > {^ -{- a2)M 

which would satisfy either the condition (2.6.18) for coming to rest in 
the discouraged state or one of the failure conditions (2.6.11) or (2.6.17). 
Similarly, either of the discouraged — > encouraged transition conditions 
(2.6.14) or (2.6.21) would imply 

(2.6.27) yK <i(3 + a2)M < {^ + ai)M 

which is sufficient for coming to rest in the encouraged state. 



34 A Mathematical Model of a Budget Control System 

The discussion to this point has been oriented towards asymptotic 
behavior. If no oscillation is allowed — since solutions which are not 
rate-stable lead to system failure — the discussion of optimal behavior 
may be restricted to systems at rest. If the department head is initially 
encouraged, and the discouragement level is moderate or extreme (or 

slight but with N < — M) then the optimal behavior and resultant rate 

of change of cost are given by 

(2.6.28) K, = l^±^iH 

7 

It = -"'^ 

If the discouragement level is shght and A^ > —M, optimal behavior 
and resultant change of cost are given by: 

(2.6.29) K, = (£±^M 

7 

dc ,7. 

It = -"^ 

independent of initial conditions. 

If the level of discouragement is moderate or extreme and the depart- 
ment head is initially discouraged, management has no satisfactor}^ 
single Kj.^ If this is the case or, in fact, if discouragement is slight and 

A^ < — M, management should initially apply a rate which is sufficientlv 

small to cause a transition to encouragement and then shift to the rate 
shown in (2.6.28). Since this second rate assures coming to rest in the 
encouraged state, the condition of nonoscillation is not violated by the 
shift in rate. 



2.7. The Oscillatory System 

It would appear at first glance that management could decrease 
the budget rapidly, causing a large reduction in cost and subsequently 
"take off the pressure," allowing the department head to recoup his 



iThe question of optimal mixed strategies in the game theory sense will not be 
explored in this thesis. 



A Mathematical Model of a Budget Control System 35 

confidence so that management can again reduce the budget at a high 
rate. However, dc/dt can never be less for any department head at any 
time than the lower of the two rates shown in (2.6.28) and (2.6.29). 
It is therefore impossible to reduce cost, even instantaneously, at a rate 
which is greater than the optimum rate obtainable with a constant rate 
of budget reduction. 

It is conceivable, however, that a greater amount of stress than M 
(say M') could be sustained for short periods while still maintaining 
encouragement, and similar remarks apply to the case for a higher failure 
point, N'. It would be necessary to keep (c — a) below M at least part 
of the time if the short-run assumption is to be maintained. Never- 
theless, a change in cost during a time interval ^o to ^i divided by the time 
interval may be expressed as 

ti — to ti — toj i^ dt ti — toj ^^ 

The right hand side of the relation is immediately recognized as — ai 
multiplied by the average stress during the interval. Hence, an increase 
in Ac/ti — to below ( — aiM) can only be accomplished through raising 
the average stress. It is commonly accepted (in psychology,^ if not in 
business management) that short periods of extreme stress must be 
counterbalanced by relatively stress-free periods of longer duration if 
neurotic behavior is to be avoided. If this psychological proposition 
holds true in the budget situation, then short periods of extreme stress 
where (c — a) > M are useful as a short run device, but in the long run 
cause a slower rate of cost reduction. 

It may be of interest, at least in passing, to examine a system which 
oscillates among four states of existence. Management will be said to be 
optimistic if it applies a rate of budget reduction which is sufficiently 
large to force an encouraged department head into discouragement and 
pessimistic if it applies a rate which is sufficiently small to cause the 
department head to return to encouragement. 

The starting point for the cycle is irrelevant, at least for purposes of 
theoretical analysis. Hence, for convenience, the analysis may be started 
with the department head encouraged but with management still 
applying the smaller, pessimistic rate which was required to restore his 
encouragement. Let it be assumed that after a time delay, ai, manage- 
ment will change its budget procedure to the higher optimistic rate of 



^See, for example, Finger (37a), Liddell (37c), especially p. 396, and Rosenzweig 
(37d) especially p. 387. 



36 A Mathematical Model of a Budget Control System 

reduction, Ka. If Ka is sufficiently large, the department head will 
become discouraged. Management will again change its tactics, and 
after a time delay 0-2, will apply the lower rate of reduction Kb, which 
should be sufficiently small so that the system will return to the starting 
point and the cycle will repeat. 

The first cycle may cause some difficulty if the department head is 
initially discouraged, and the discouragement is extreme. In order to 
allow management to choose a pessimistic rate for the oscillatory system 
which is sufficiently small to return the department head to encourage- 
ment in the steady state oscillation, but which may not be small enough 
to encourage the department head initially if the initial stress is very 
large, a third rate Kc is introduced which can be used initially, but 
ignored after the first management decision to change the budget. 
Otherwise, the rate Kb can be applied initially without complication. 

After the first cycle, the discrepancy at the start of the cycle will be 
M since the department head will have just become encouraged. For 
the encouragement — > discouragement and discouragement -^ encourage- 
ment transitions, respectively, the budgeted rates of cost reduction Ka 
and Kb required to maintain the cycle are specified in the ''management 
action" column of Table 2.1. The expressions for the discrepancy c — a 
as a function of time are shown in Table 2.2, which follows. The deriva- 
tions are a result of straightforward but tedious application of equations 
(2.5.6a), (2.5.6b), (2.6.9), and (2.6. 16) 1. The change in cost which occurs 
during a cycle is 

(2.7.2) Ac = - -^ [Ka(ti - cTi) + KBa,] 

IKb(t2 — (T2) + Ka(T2] 



-f- Oi: 



where n and t2, as shown in Table 2.2, are the amounts of time spent in 
the encouraged and discouraged states, respectively, and their sum is 
the cycle time. Since both the cycle times and the cost reduction during 
a cycle are independent of both the absolute level of costs and (after 
the first cycle) of initial conditions, the oscillatory system is equivalent 
in effectiveness to a system with a constant rate of cost reduction given 

by 

(2.7.3) ^^ = -^ 

at Ti + T2 



^These derivations, as well as a rigorous proof of the inferiorit}' of the oscillatory^ 
system to the steady state system as a cost minimization device, are contained in 
the author's earlier paper (75), pp. 26-42. 



A Mathematical Model of a Budget Control System 



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38 A Mathematical Model of a Budget Control System 

The admissibility criterion for the steady-state model may be replaced 

by 

(2.7.4) Ac < 

The existence of admissible solutions is assured. Both ri — ci and 
T2 — <Ji are positive, but the former can be made as large as desired by 
allowing '^Ka to approach (j3 + a-^M while the latter can be made to 
approach as close to zero as desired by reducing Kb^ 

2.8. A Technological Constraint 

In the interest of emphasizing the motivational aspects of budget 
control and not unduly complicating the model, technological constraints 
have been ignored. A technological constraint can be readily imposed, 
however, by the alteration of equation (2.5.2), leaving the model other- 
wise unchanged. Assume that, for a given level of stress and state of 
encouragement, it becomes increasingly harder to reduce costs as c 
approaches an optimum value, c} This assumption can be comprehended 
in a relation of the form 

(2.8.1) -J = -ale - a)e-^''-^ where c = c{t) 

and where c and c are defined at some output level for a given ''tech- 
nology." 

Note that questions of discrepancy between budgeted, aspired, and 
actual outputs are not dealt with as explicitly related to the technological 
constraint; only the rate of cost change is directly affected. 

The adjustment of the aspiration level, viz. 

(2.5.1) | = ^(-«)+^S • 

remains unchanged. Provided conditions analogous to those for admis- 
sible rest in the original model/ 



Uhid., p. 34. 

2Cf. Charnes and Cooper (12) who discuss, in addition, the adjustment of c for 
output rates. 

^Given an initially encouraged department head, or a budget sufficiently high to 
cause encouragement, thus resolving initial conditions difficulties, followed by a shift 
to the budget shown in (2.8.2a). Similarly, if c — a is initially negative, application 
of rates of budget reduction as constrained in (2.8.2) will clearlj^ force c — a to 
become positive or zero. 



A Mathematical Model of a Budget Control System 39 

(2.8.2a) < - 7 ^ < M{^ + aie-^f'^-^) or, 0:2 > 

(2.8.2b) N(I3 + a2e-'''-^) > - yj> M(/3 + cxiC-^''^-^) 

are met, < c — a < M. Ifc — a = and 7— = the system is at 

CLl 

dc 
equilibrium. If c — a ?^ 0, then -r: < and c—^c. But 

CiL 

(2.8.3) lim ^ = 

Hence the revised model is stable with an equilibrium solution given by 

(2.8.4) «.T ^ e-^/-^ = 

which requires, if ai, 7 > 0, either a zero rate of budget reduction or 
c = c. The optimum rates of budget reduction are revisions of equations 
(2.6.28) and (2.6.29), viz. 

db ^ M{I3 + aie-'f'-^) 

(2.8.5) ^^ ^ 

unless discouragement is slight and N > — M, in which case 

(2.8.6) "^^ '^ 

The model may be considered as a linear model with a finite number 
of discontinuities by letting 

(2.8.7) wr = e" ''r-c where Cr < c < Cr + p 
which provides 

(2.8.8) . where Cr < c < Cr -\- p 
Hm cor = e ^'"r c 

p— >o 

allowing management to apply the rate of budget reduction 

(2.8.9) K, = m±^^ or K^ = ^±^2!^ 



40 A Mathematical Model of a Budget Control System 

depending upon whether it is desired to approximate (2.8.-i) or (2.8.5), 
which will produce a rate of cost reduction less than the optimal rate 
but may be made to approach it by increasing the frequency of decision. 

2.9. Conclusions 

To comprehend the more complex situations which are hkely to be 
encountered in practice it would undoubtedly be necessar\^ to comphcate 
the model, and this would, in turn, vitiate the objective of simplicity 
and clarity which is a sine qua -non of a theoretical formulation at this 
stage of scientific work in budget control and cost behavior. But simple 
as it is, the analysis here presented does present some highh^ plausible 
clarification. For instance, once the department head's goal-setting 
pattern has been established, a static budget vnH tend to produce 
stationary expected cost, subject onh' to random variation about an 
expected value, in a viable ongoing situation. It is evident, furthermore 
(and psychological considerations appear to be strong enough to warrant 
this conclusion), that management cannot choose a rate of budget 
reduction for a particular department independent of considerations of 
the motivation structure of the department's head, .\lthough this would 
appear to be obvious, the emphasis in today's hterature is on budgeted 
costs and their relation to technolog}' and not on their relationship to 
the individual being budgeted.^ As noted in Section 2.8, technological 
constraints are an additional factor which must be considered, but 
technology.- important though it may be. does not ob^'iate the necessity' 
of also considering m an\' measure motives in a budgetary- system which 
ultimateh^ depend on some real consensus for their implementation. 
It is paradoxical that those who criticize mechanistic approaches to 
accounting (e.g., the research which treats human beings as ser\'0- 



^E.g., Keller (40\ p. 98, states that, '•The setting of standards is the responsibility 
of the technical staffs of a plant such as industrial engineers, design engineers, and 
chemists," although he later concedes that the foreman must agree that the standard 
is '"fair." The problem of what to do if agreement is lacking, or what proportion 
of standards should be made to come into the area of questionable '"fairness," is not 
related to the motivations of the individual concerned. 

2In the literature of theoretical economics such technological factors (in the form 
of a production function) are accorded preponderant importance. But it must be 
remembered that the economic theory of the firm is based on a highly simplified model 
of the firm's "human" structure which, in turn, is justified by the fact that this model 
is designed primarily for analyzing "market" or general economic behavior and not 
the beha\'ior of agents within a single firm. 



A Mathematical Model of a Budget Control System 4«1 

mechanisms^ fall into the trap of a mechanistic approach themselves 
when applying (or explaining) rules of thumb to the problems of budget- 
ary control. 

The mathematics used in this chapter (and the logic with which it is 
associated) has been directed primarily to laying bare (and clarifying) 
certain issues which, though sometimes recognized in practice, are often 
concealed — or go completely unattended — in the existing literature on 
budgeting. 2 Certain by-products have also been achieved which will be 
explored in various ways in the chapters that follow. Thus certain issues 
involved in the strategy of setting budgets have been uncovered and 
related to each other in a way which related certain major factors to one 
another. Thus the budgeted amounts of the person to be controlled have 
been related to his actual cost performance with the aspiration levels 
acting as an intervening variable. Moreover, possible interactions 
between aspiration levels and cost performance have been examined. 
Finally, the objectives of those who seek to influence cost performance 
have been brought into the analysis, via the budgetary variables, in a 
way which raises most issues of strategy relative to the objectives of 
central management and thereby brings to the fore certain questions 
which are germane to adequate performance of the controller's office, 
as that office is now conceived.^ 

It is obvious, for example, that blanket budget reductions which are 
common in government bureaus and similar cost-saving "drives" in 
large corporations on a plant-wide scale are of dubious merit. Further- 
more, the treatment of all subordinates ''impartially" when it comes to 
budget demands, which essentially means treating them equally, regard- 
less of their motivation structure, appears not only irrational from the 
cost standpoint but from the standpoint of welfare of the subordinates 
as well. For example, if a man at middle management level is directed 
to cut his budget, he may be able to "push" one man whose discourage- 
ment point is high to the limit and by so doing avoid discouraging a few 



^See, for example, Anthony (1), who (rightfully) states that, "Human control 
systems cannot be so easily or so precisely designed as mechanical or electrical ones." 
Although he rejects the servomechanical analogy, he can only offer in substitution 
such comments as,"The method of constructing costs for control purposes is governed 
by management policy." 

2Cf. e.g., Heckert (34) or MacDonald (53). This literature is almost completely 
occupied with the mechanics of budgeting to the point where it has assumed an 
almost standard form of presentation and development. 

^Particular reference is made to the "control," as distinct from the "service," 
function in the sense in which these two terms are used in the controUership literature, 



42 A Mathematical Model of a Budget Control System 

others whose discouragement points are lower, thus preserving morale 
and reducing costs further than he could by behaving ''impartially." 
More specifically, if management desires to behave consistently over 
time, it must choose a more modest rate of budget reduction for the man 
who is easily discouraged than for the man who appears perpetually 
enthusiastic. Given two men with the same discouragement point, 
management must avoid discouraging the man who 'Svhen he is bad, is 
horrid," whereas the man who 'Vhen he is bad, is still shghtly good" 
and doesn't give up easily can be kept on the verge of resignation for best 
cost results. 

The models explored in this chapter indicate that an increase in stress, 
up to a point, is desirable in the reduction of costs. The assumption that 
standard costs must be ''attainable," which pervades the current budget 
literature, is based on the assumption that the people who operate under 
them must be satisfied if they are to turn out a reasonable but unexcep- 
tional performance. But under even the very simple assumptions of the 
models in this chapter, it is evident that this need not be the case. Under 
certain circumstances the cost expected to obtain by a department head 
must be above his aspirations in order to ensure that he will work 
diligently toward reducing costs. Insofar as budgets affect his aspirations 
this kind of behavior must be taken into account; references to "loose" 
and "tight" budgets, with blanket approvals of the latter and condem- 
nation of the former, as is common in the literature,^ are not an adequate 
basis for dealing with this problem. Pending an explicit quantitative 
characterization in particular circumstances, the equations used in the 
models of this chapter have at least established a provisional quahtative 
characterization which suggests, instead, that budgets should be set 
rather in a way which allows an affected department manager to achieve 
his aspirations part of the time. In conclusion we note that this opens a 
rather broad range of questions concerned with the value of accuracy 
and timeliness of accounting (as distinct from budgeting) reports in terms 
of both their immediate and ultimate consequences for cost behavior. 



^Some authors entirely dodge this issue by favoring only "accurate" budgets, 
failing to make clear whether they are speaking of the budget as a planning instrument 
or a control instrument. 



CHAPTER 3 



Available Empirical Information 



3.1. Introduction 

If, as is sometimes said, "Control exists in the minds of men rather 
than the books of account," then the theoretical model of Chapter 2 
helps to highlight and formalize the kinds of psychological concepts that 
need to be considered. Of course such formalisms, however logical or 
elegant, are not enough in and of themselves to justify a theory. An 
empirical foundation is greatly to be desired or, failing this, some kind 
of testing and validation is required. Apart from the study by Argyris 
(2), which does not really deal with the central problem of cost responses 
to budgetary control procedures, there is (unfortunately) no systematic 
accumulation of evidence where the desired empirical foundation can 
be readily secured. 

Failing access to a broadly based and systematic series of studies of 
managerial behavior under different budgeting arrangements, the 
following seem to be the best immediate alternative sources of empirical 
information: (1) studies that have been made of worker reactions to 
various incentive pay schemes, and (2) psychological (laboratory) studies 
(e.g., in aspiration level theory) which are more or less germane to the 
topic of interest. After these topics have been discussed in the sections 
immediately following, attention may be turned to one other source 
of information and possible validation. This information will be reported 
in the form of a laboratory experiment designed explicitly for the purpose 
of testing salient aspects of the theory which has now been advanced. 

3.2. Laboratory Experiments on the Level of Aspiration 

In this area, work in the laboratory has concentrated on descriptions 
of the formation of goals, taking performance as the independent 

43 



44 Available Empirical Information 

variable, rather than manipulating performance through the effect of 
aspirations and external goals. However, valuable supporting evidence 
for the postulates of the mathematical model regarding aspiration level 
formation is available. 

Chapman and Volkmann (7) asked a group of 86 students in psy- 
chology classes what score they hoped to receive on a test of literary 
achievement in which scores could range from 17 (by a random choice 
of answers) to the maximum of 50 questions. The students were divided 
into four groups. Group A was given no additional instructions while 
Group B was told that an average score of 37.2 was obtained by a group 
of authors and literary critics. Group C the same score for a group of 
students in psychology, and Group D the same score for a group of 
"unselected WPA workers." 

The average aspiration levels^ chosen by the four groups were 
26.95 d= 6.33, 23.09 d= 3.46, 31.09 =b 8.95, and 33.05 db 8.57. These 
differences were significant in pairs at better than the 1 per cent level, 
except for the A and B difference and the C and D difference, in which 
the probabilities of significant difference were .962 and .767, respectively. 
Also of interest was the significant correlation (.523 d= .105) between 
aspiration level and performance in Group A. The authors concluded 
that, "The level of aspiration estimated in advance of performance is 
estimated neither at random nor without reference to the ability to 
perform the task."^ The authors interpreted the results as evidence of 
the influence of the social environment in goal formation. It seems 
unlikely, however, that a group of college students would aspire to 
(i.e., set a personal goal at) a score 4 points lower than that attained by 
"an unselected group of WPA workers," although it is conceivable that 
their performance estimate^ might incorporate such indications of 
inferiority feelings, interpreted perhaps in terms of fear of a new task. 

Although the data show differences caused by reference to social 



iJt should be noted that the definition of "aspiration level" is not wholly' consist- 
ent among writers in the psychological literature. For instance, the definition used 
by Chapman and Volkmann is not consistent with that of Frank (33), in that the 
latter explicitly requires previous experience with the task and makes formal and 
explicit provision for this, whereas this is not the case in the stud}^ reported above. 
In fact Chapman and Volkmann believe that the definition of aspiration level should 
be enlarged to include all estimates of performance that might be offered b}' the 
subjects regardless of previous experience. 

^Chapman and Volkmann (7), p. 283. 

^Unfortunately, I have not found a completely satisfactory method of separating 
aspiration and performance estimates, either. A discussion of this problem appears 
in Chapter 4. 



Available Empirical Information 45 

groups, they also indicate that the subjects did not aspire to group norms 
in the abstract. For instance, if strong conformity to group norms were 
to be predicted, then Group C should have had an aspiration level 
average of about 37 instead of 31, since this score, the only general clue 
given them, explicitly referred to **a group of students in psychology." 
Moreover, on this hypothesis this group should have had a smaller 
variance (regarded as a measure of heterogeneity) than any of the other 
groups; whereas, in fact, the variance of this group was larger than any 
of the other three. An alternate explanation of the findings might be the 
perception of the 37.2 figure by the subjects as an external goal rather 
than a norm of a group, and with reference to this goal a subject would 
set his aspirations according to his perceived relationship to the group. 
But, of course, even this hypothesis is not fully satisfactory unless 
buttressed by further explanations such as the fact that the students 
were accustomed to aspiring to (or more likely predicting) performance 
below an externally imposed goal, while the amount of goal mitigation 
which took place was significantly affected by the reference to social 
groups. This conclusion would provide evidence for the validity of 
postulate (i) of the mathematical model; i.e., that the aspiration level is 
not necessarily set equal to the external goal, but is affected by it. 
A second experiment of Chapman and Volkmann (7) was designed 
to assess the effect on level of aspiration of the achievements of other 
groups in a situation where they possessed prior knowledge of their 
performance. Four forms of 32 items each, taken from the Otis Self- 
Administering Tests of Mental Ability, were administered to the subjects 
on four consecutive days. On the second day the score on the first day's 
test was presented to the subjects and their aspiration level attained 
before beginning the second test. On the third day, the subjects were 
split into two groups matched in performance on the second day's test. 
Each subject in Group A was given the previous two days' scores and a 
third number (which was actually the average of his own test scores 
amended by d=.9 to camouflage it) which he was told was the average 
score attained by a group of ''unselected WPA workers," while a Group 
B subject was given the same information for "New York members of 
the National Academy of Sciences;" all subjects were given their scores 
and aspiration levels were ascertained. On the fourth day, each subject 
in Group A was told his score of the previous day as well as, ''The 
average score of the class to date is 5.2 points below your average score to 
date."^ The same information was also supplied to Group B except 



K7), p. 235. 



46 Available Empirical Information 

that each of these persons was told that the average class score was 5.2 
points ''above" his average score to date. 

The test results then gave estimated probabilities of a true difference 
between the mean aspiration levels of the two groups on the second, 
third, and fourth days as, respectively, .516, .655, and .520. The authors 
concluded that, "Under the conditions of this experiment, w^hich included 
prior performance and knowledge of this performance, the level of 
aspiration was not changed by knowledge of the achievements of other 
groups." 1 These results would seem to indicate that in budgetary 
practice, the aspiration levels formed relative to similar budgets in 
several departments might be considered independently, particularly if 
the "my department is unique" attitude were initially prevalent. On 
the other hand, the experiment casts serious doubt on the possibility 
of affecting the aspirations of one department head by pointing out 
the achievements of another. 

Other experimenters have concentrated on the generalization of 
expectancy along need-related lines. ^ CrandalP gave subjects two sets 
of stories similar to the Thematic Aperception Test. The experimental 
group was given a motor coordination task between the two sets, and 
told they did badly. It was found that the second set of tests showed 
lower "freedom of movement,"^ relative to a control group which rested 
between sets of stories, in the areas of physical skill, academic skill, and 
love and affection of opposite sex peers. The differences between the 
groups were reported as significant at the 1, 2, and 20 per cent levels, 
respectively. The size and gradient of the decrements in freedom of 
movement (expectancy) were interpretated as indicating that the failure 
in motor coordination was generalized to need-related areas to an extent 
that was dictated by the closeness of the area to the need for recognition 
of the skill in which the failure experience occurred. If the results were 
to be taken at face value, the implications for budget control would be 

Ubid. 

2The remarks offered on these experiments are based entirely on the work reported 
in Rotter (67). A good deal of additional work has also been done in the form of a 
series of Ph.D. dissertations at Ohio State University by S. J. Dean, R. L. Dunlap, 
A. A. Lasko, and D. E. Hunt. Unfortunately, however, I have yet to secure access 
to these additional documents, so the comments in this section rest entirely on 
Professor Rotter's reports including his summaries of the results secured by Drs. 
Chance, Crandall, and Jessor. 

3(22) cited in Rotter (67), pp. 120-122. 

^Generally speaking, a measure of expectancy. In essence, the expectancy or 
performance estimate in an area appears to be correlated with the "abandon" with 
which a subject deals with that area. See (67), p. 110. 



Available Empirical Information 47 

that no standard or budget for an individual could be set without 
reference to all of the others, since the need for recognition of skills 
required for budget attainment could not be expected to vary substan- 
tially from one budget to another. This need-relatedness would require, 
in addition to the difficulties of programming to determine the desired 
performance in each cost or performance subgroup, additional program- 
ming to take into account the effects of success or failure in achieving 
one budget on expectancy (aspiration) of another, as well as the main 
effect. However, as Rotter points out, "The results of this study 
could be explained by other theoretical approaches or as a function of 
uncontrolled similar factors in the experimental design."^ 

In an experiment of R. Jessor,^ subjects were asked their expectations 
and minimum goals with regard to their performance on tests of (1) 
arithmetic, (2) vocabulary, (3) motor skill, and (4) social skill on which 
the average score was 25, ranging from 0—50. After the subjects were 
given an arithmetic test, they were told that their scores were a pre- 
determined amount either below or above their minimum goal on that 
test. They were then asked to re-estimate their scores for another form 
of the arithmetic test and, since they had not taken the other three tests 
yet, allowed to change their estimates on the other three tests. The 
results were reported in terms of the proportion of subjects changing 
expectancies and minimum goals. Approximately 90 per cent of the 
subjects changed expectancies on the arithmetic task, 45 per cent on 
the vocabulary task, 35 per cent on the motor skill task, and 30 per cent 
on the social skill task. The proportions changing minimum goals were 
lower, but followed a similar gradient; approximately 65, 35, 25, and 20 
per cent for the same four tasks. The gradients indicated were predicted 
by the experimenter on the basis of the amount of similarity of need 
or reinforcement based on a priori estimates of the similarity. It is my 
impression (borne out by experience with the experiment to be described 
in Chapter 4) that the ego-involvement of subjects in laboratory ex- 
periments, particularly those conducted in a classroom situation, is 
frequently exaggerated. The change of expectancies of Jessor's subjects 
could have been merely a result of the subjects ''testing the test." 
Having the ability to utilize common sense along with the experimenter, 
a subject might feel that the information which he derived from his 
experience on the first test would be highly relevant to his estimation 
of the difficulty of a similar test. He may perceive the information he 



Ubid., p. 121. 

2(39), in Rotter (67), pp. 121-124, 169, 191, 214, 215, 326. 



48 Available Empirical Information 

possesses to be relevant to assessing a paper test of verbal ability, but 
unlikely to aid him in determining the scoring system to be used in a five- 
minute interview with a female member of the staff (test of social skill). 

It would appear that the four tasks also follow a gradient in terms of 
testability; it is likely that this gradient would be perceived by the 
subjects. Furthermore, the gradient follows an introspective notion of 
the probable willingness of the subjects to change expectancies and 
minimum goals, in terms of the degree of relationship perceived by the 
subjects between the task and the Gestalt. I would predict, on this basis, 
that if the subjects in fact accepted the test score as a measure of their 
abilities, performance of the experiment in reverse (with the social skill 
task used as the reference point) would produce a considerably reduced 
gradient. On the other hand, the hypothesis that expectancy generaliza- 
tion is based on similarity of need would necessarily lead to a prediction 
of a new gradient very similar to the original. The changes in minimum 
goals could be interpreted as following directly from the changes of 
expectancy in order to (1) avoid the logical inconsistency of a minimum 
goal greater than the expectation and/or (2) maintain a more or less con- 
sistent difference between the expectancy and minimum goal perceived 
by the subject as "proper." Thus the experiment does show generaliza- 
tion of expectancy, but it falls short of demonstrating the dependence 
of generalization upon similarity of need. 

A third study, which was designed to eliminate some of the physical 
stimulus generalization of the studies of Crandall and Jessor, was per- 
formed by Chance.^ The tests were the same for the four experimental 
groups and of an unstructured nature based on more or less standard 
versions of ink blot and word associations tests. The four groups were 
told, respectively, that (1) both tests measured heterosexual adjustment, 
(2) both tests measured leadership potential, (3) the first test measured 
leadership potential and the second heterosexual adjustment, and (4) the 
first test measured heterosexual adjustment and the second leadership 
potential. The subjects stated expectancies for both tasks, performed 
the ink blot test and were given predetermined scores of either 7 or 14 
points above their expectancies on the first task, and then were asked 
to restate their expectancies, which they could change if they wished. To 
quote Dr. Chance, she found that, "When the two tasks were described 
as need-related, there was significantly greater generalization than when 
they were described as measuring different skills. "^ Unfortunately the ex- 



iChance (6) cited in Rotter (67), pp. 124-125. 
2Rotter (67), p. 124. 



Available Empirical Information 49 

periment as it is described need not show anything about need similarity, 
but shows rather that subjects are hkely to think that two tests measur- 
ing the same skill are likely to have the same scores. Perhaps the more 
important conclusion of the experiment is the indication that the 14- 
point increase resulted in significantly greater ''generalization" than the 
7-point increase; it would tend to indicate that the subjects have some 
intuitive notion of the concept of "significant difference" and the 7-point 
difference was not sufficiently far from the original value to merit 
changing the expectancy.^ Either a 7- or a 14-point difference might 
seem insufficient motivation for inducing search behavior in some 
subjects but sufficient in others. 

A possible lacuna in the three experiments, as reported, is that the 
results did not include the possibility that the change in expectancy was 
a result of the subjects' re-evaluation of the tests relative to their self- 
conception (which remained unchanged) rather than as a result of re- 
evaluation of themselves in terms of the tests. It is certainly conceivable 
that the adjustive reactions to even repeated failures on examinations 
for a population of students might, with some exceptions, tend to be 
extrapunitive.2 That is, their covert (and possibly overt) behavior might 
tend towards re-evaluation of the test, the difficulty of the course, the 
fairness of the instructor, and the merit of the institution, at least for 
some period prior to undertaking an initial introspective appraisal. If 
these observations are true then some cloud may be cast on a series of 
studies which presuppose, by assumption only, that a subject's self- 
conception is changed radically during the course of an experiment 
involving one or two test successes or failures. 

An aspiration level experiment which attempted to increase ego 
involvement by taking the subjects at least partially out of the labora- 
tory was conducted by Siegel (70). A class of students in statistics agreed 
to accept a semester grade on the basis of an interview with an exper- 
imenter other than the professor. The professor was to return in an hour 
to discuss a student's grade if he were dissatisfied. All students were 
given the grade of C. 

The students who wished to discuss their grades, by and large, were 



^A hypothesis of Simon might be recalled here — viz., that a search for a new goal 
will not be instituted unless the original goal becomes unsatisfactory. See Charnes 
and Cooper (10), and see Chapter 2, p. 21. 

^See (37d) for a more complete definition of this term; generally speaking, it is 
intended to describe cases in which an individual aggressively attributes his frustra- 
tion or failure to external persons or causes rather than to deficiencies arising from 
his own being or behavior. 



50 Available Empirical Information 

those who would have received an A or B in the course by the normal 
grading scheme.^ Siegel concluded that the ones who returned did so 
because their aspiration level exceeded their grade. However, the grade 
has intrinsic reward value in addition to its more obvious function as a 
measure of achievement. It is not only that a student who aspired to 
an A received a C (he may have expected an A, in which case his ability 
to estimate his abilities was in question; or he may have thought he 
deserved^ an A which involves principles of fairness, reward for hard work, 
etc.). Whatever its other merits, however, the Siegel experiment does 
not give results which could be an aid in the topics with which this book 
is concerned, except possibly to underscore the need for carefully defining 
pertinent characteristics of the variable under consideration. 

A rather complete survey of the field of aspiration level experimenta- 
tion up to 1944 has been performed by Lewin, Dembo, Festinger, and 
Sears (37b), which, with the exception of the first Chapman and Volk- 
mann experiment, contains empirical studies which have not been treated 
here. The experiments they discuss have generally dealt with the precise 
interpretation of ''level of aspiration" in various contexts and causal 
factors in shifts of the aspiration level. The causal factors investigated 
include social and personality factors as well as temporary situational 
factors. The theoretical model which these authors present to "clarify 
a situation which is at present a bit chaotic and to give orientation to 
further experimentation"^ has been discussed in Chapter 2. In view of 
the situational dependence apparent in these studies, and the general 
lack of attempts to assess the possibility of influencing the behavior of 
the subjects through the aspiration level, nothing but their most general 
conclusions are possibly applicable to the problems of budget control as 
defined in this thesis. Since these general conclusions have alread}^ been 
stated (during the course of discussing other experiments), there is little 
point in singling them out for further separate treatment. 

3.3. Experiments on the Behavior of Animals 

A great deal of experimental psychology has been devoted to studying 
the behavior of animals. The advantages and limitations of this ap- 



iThere were some few exceptions to this statement. But these cases could have 
been predicted by previous experience with these individuals. 

2E.g., via experience with past class norms, knowledge of course performance by 
his fellow students, etc. 

3(37b), p. 356. 



Available Empirical Information 51 

proach are well-known, with the latter being particularly acute in any 
study of budgeting applied to managerial behavior. Nevertheless, in the 
interest of more complete documentation it may be well to make a few 
summary references to these results. 

A case in point is the study of Birch (4). He tested six young chim- 
panzees and found that they performed best in three substantially 
different types of problems after a food deprivation of 6 hours, with food 
as the reward for a correct solution. The two food deprivations involving 
lower motivation (2 hours and 12 hours) ^ produced substantially poorer 
results, although the effect was more marked in those problems for which 
time of solution was the important variable. Among the more highly 
motivated circumstances, the 24-hour deprivation caused only slightly 
poorer behavior than the 6-hour trials, while the 36- and 48-hour 
deprivations produced markedly poorer performance. The percentage 
of ''insightful" solutions to a hooked rope problem and 10 problems in the 
use of sticks was greatest for the 24-hour deprivation, and in descending 
order, 12, 6, 36, 2, 48. 

Although the sample is too small to be reliable for this variable, it is 
interesting to note that the satisfied chimpanzees did not perform more 
insightfully when they had ample time to think about their problems. 
On the contrary, the best performance from the standpoint of insight 
appeared at a food deprivation (stress) somewhat greater than that of 
the best level of effective performance. In the 36- and 48-hour trials 
stereotypy was common and, for what it may be worth, the overall 
results are quite consistent with postulates (iiia) and (iiib) of the mathe- 
matical model in the preceding chapter. 

Another aspect of behavior which may be of relevance to budgetary 
practice is the motivating effect of fear. This aspect has been extensively 
investigated by Miller (78a) in a series of experiments on rats. Of 
particular interest is one in which learning behavior was tested after 
the initial pain stimulus had disappeared. 

Each rat was given 20 shock trials, mixed with a larger number of 
nonshock trials, in one compartment of a two-compartment box. The 
rats quickly learned that they could escape through an open door into 
the second compartment. When placed into the same compartment 
without shock but with the door closed, the rats learned to open the door 
by pressing a lever. The maximum speed of door opening was attained 
by the rats between 5 and 9 days after the shocks ceased ; a period which 
involved 100-180 nonshock trials. After a period of 30 days (360 non- 



^The 12-hour is pre-breakfast whereas the 6-hour deprivation is a missed dinner. 



52 Available Empirical Information 

shock trials) some of the rats were still pressing the lever to "escape" 
from the compartment in which the danger of shock had been long 
nonexistent. 

Miller states that, "We may venture the hypothesis that learned 
anticipatory drive-reducing responses of fear obscure the true rate of 
drive in smoothly functioning human behavior." ^ If this hypothesis were 
to be substantiated, the effect of fear of dismissal or fear of demotion 
might produce (assuming the current state of knowledge) unpredictable 
responses to the budget control mechanism. Though beyond the scope of 
this paper, it is reasonable to assume that these unpredictable responses 
are likely to run counter to organization goals, and hence investigation 
of their effects would seem advisable in future research. 

Still a third factor which would relate to the rewards associated with 
budget attainment has been investigated by Wolfe (85) who tested the 
effect of delay in reward on the performance of chimpanzees. He found 
that a group that, immediately following a work task, was given tokens 
with which to buy food at the end of a delay period was more willing to 
work than a group that performed the work task and merely waited for 
the food. The first group could tolerate a longer delay than the second, 
but when the delay became sufficiently long, the group could not be 
induced to work for the token. 

He also tested the effect of the "capital stock" on the willingness 
to work. The chimpanzees were allowed to work for ten minutes to 
"earn" tokens with which to buy food. When they had no tokens at 
the beginning of a trial they earned an average of 21.2 tokens. ^\Tien 
given 5, 15, and 30 tokens at the beginning of a trial, they only worked 
hard enough to earn an average of 15.2, 4.2, and 2.6 tokens, respectively. 

Wolfe's experiments indicate that the amount of reward is not the 
only factor which influences performance. On the contrary, the timing 
of reward, the way it is administered, and the amount of reward on hand 
prior to the performance are factors which must be considered. 

3.4. Studies of Utility Measurement 

Another field which might be pertinent is the area of work which is 
concerned with empirical investigations in the measurement of individual 
utilities. This field is of interest at least insofar as it casts light on the 
problem of influencing behavior by means of monetary reward. For 
this reason some results of experiments in utility measurement will be 
presented. 



iMiller (78a), p. 451. 



Available Empirical Information 53 

Davidson, Suppes, and Siegel (24) performed gambling experiments 
with 19 subjects. The subjects were asked to choose between two 
alternative gambles, each of which had a small positive or negative 
payoff depending upon the throw of a die. From these experiments 
they concluded that:^ 

1. Under controlled conditions, some people (15 out of 19 subjects 
in the present experiment) make choices among risky alternatives as if 
they were attempting to maximize expected utility even when they do 
not make choices in accord with actuarial values. 

2. For such people it is possible to construct a utility curve unique 
up to a linear transformation . . .^ 

3. Of the 15 subjects whose utility curves were determined, 12 had 
curves which were not linear in money. 

4. Some evidence was obtained for the secular stability of subjects' 
utility curves . . . 

Mosteller and Nogee (63) computed utilities of money (in small 
amounts), using 15 subjects in a gambling situation. In the basic game, 
the subjects were allowed to choose whether or not they wished to bet 
5 cents on beating a particular "hand" in poker-dice. There were seven 
hands whose odds were such that a ''fair offer" varied from 2j/^ cents to 
5 dollars against 5 cents. The utilities, expressed as a multiple of 5 cents, 
were computed in terms of the bet which the subject was willing to make 
50 per cent of the time (interpolating where necessary). Thus if a subject 
were indifferent to risking 5 cents to win 35 cents on a bet whose odds 
were 10:1 (fair offer = 50 cents), his utility of 35 cents would be 10 utiles. 
The subjects were given information as to what was a ''fair offer" for 
each hand before the trials which were used for computation of utilities, 
and hence were aware of the objective probabilities and thus might have 
maximized expected return, had they assumed the experimenter was 
benevolent. It is, perhaps, not surprising that they did not attempt to 
do so. Three groups of subjects, run in groups of five, indicated con- 
siderably more similarity of utility within groups than between groups. 
Harvard graduates comprised two groups, while low income National 
Guardsmen comprised the third group. The Harvard students appeared 
to have a definite decreasing utility of money — but is it possible that 
they really wished to appear "conservative" before their fellow stu- 



1(24), p. 80-81. 

2As required by the von Neumann-Morgenstern axiomatization of cardinal utility. 



54 Available Empirical Information 

dents? ^ The Guardsmen, on the other hand, appeared to have increasing 
utiUty of money, but this could be explained purely in terms of gambling. 
Anyone experienced in games of chance is aware of the importance of 
being "a sport," and the Guardsmen, probably more experienced and 
having a higher utility of gambling per se, may very likely have vied with 
one another for their willingness to take risks since it was relatively 
inexpensive. Thus, the utility measurements of this experiment are 
very difficult to relate to a nongambling situation. 

In addition to the problem of the utility of gambling, W. Edwards 
(30) has shown that subjects have a particular preference for certain 
probabilities. Furthermore, he found that subjects tend to exhibit 
greater preference for ''long shots" when they are actually playing 
for money than when they were playing for worthless chips or "just 
imagining." Perhaps the most valuable piece of information that can 
be gleaned from the gambling literature, applicable to budget control, 
is the following, which is derived from observations on subjects who 
were forced to choose among bets whose expected value was negative 
( — 52^ cents for all). As Edwards states, ^ 

The dominant fact about NEV [negative expected value] bets is that 
S's don't like to lose — they rather consistently prefer the alternative 
which had the lower probabilities of losing (and of course the higher 
amount of loss). This trend is so strong . . . that it obscures the other 
relationships which may be present. 

If this result could be extrapolated with confidence to a typical 
budget situation, say the case of a department head who has several 
budgets to "make," then assuming that the penalty for missing each is 
the same, a rather interesting conclusion would emerge. The department 
head would, on this hypothesis, choose a course of action w^hich would 
reduce the likelihood of ''missing" on any budget instead of making all 
but one "safe" at the expense of the last. Such information would be 
extremely valuable as a basis for rational budgeting even after allowances 
were made for the fact that only an "average" or "representative" 
department head might behave in this manner. 



iTwo subjects in one group refused to bet more than 50 per cent of the time on an 
offer of 10 dollars where a fair offer was 5 dollars. The authors indicate that "for 
some subjects, the amount of money required to induce play against a hand may be so 
large that a project such as this one cannot afford the information." ((63) p. 382.) 
Since they stopped at an offer whose expected value was 10 cents, which would hardly 
seem exorbitant, it is probable that the authors were thinking in terms of absolute 
rather than expected value. 

2(30), p. 359. 



Available Empirical Information 55 

Unfortunately, the case is not so clear either in logic or in evidence. 
J. Dreze (27) has presented a model designed to allow any decision 
situation to be described in terms of complex games. He indicates, 
however, referring to the experiments noted above and others utilizing 
similar monetary amounts, that "It seems highly dubious that obser- 
vations about such trivial decisions could by extrapolation throw any 
Hght on the behavior of similar persons when faced with truly important 
problems. ''1 

However, even the case for rationality is not uniformly accepted, as 
the following quotations show. For instance, Professor J. Marschak 
argues persuasively for the usefulness of rationality as a basis for ana- 
lyzing industrial decision-making along the following lines i^ 

At this point we can only hint at what is probably the most impor- 
tant virtue of the advice to maximize expected utility and, hence of 
the behavior postulates implied in this advice. We conjecture that, 
for a large class of distribution functions and utility functions, the 
following proposition is true : if every action is chosen in such a way as 
to maximize the expected utility, then, as the number of such actions 
is increased, the probability that the achieved utility differs from the 
maximum utility by an arbitrarily small number, approaches unity. 

On the other hand, W. Edwards has presented a rather convincing 
opposing view. He first notes that :^ 

The crucial fact about economic man is that he is rational. This 
means two things: He can weakly order the states into which he can 
get, and makes his choices so as to maximize something. 

and, in later discussion:^ 

There has, incidentally, been almost no discussion of the possibility 
that the two parts of the concept of rationality might conflict. It is 
conceivable, for example, that it might be costly in effort (and therefore 
in negative utility) to maintain a weakly ordered preference field. 
Under such conditions would it be rational to have such a field? 

A recent study of business decision-making conducted by Cyert, 
Dill, and March (23) presents some evidence for questioning the concept 
of ubiquitous rationality in the behavior of businessmen. The question 



iDreze (27), p. 48. 
2(59), p. 139. 
3(31), p. 381. 
^Ibid., p. 382. 



56 Available Empirical Information 

of whether or not business should behave rationally — and if so, where 
rational behavior might be ''rationally" justified — is discussed elsewhere 
in this thesis. For the sake of concreteness this discussion is focused on 
specific types of decisions. However, in the context of attempting to 
assess the aspects of human behavior which would be useful in deter- 
mining the reaction to budgets under various reinforcement schemes, the 
assumption of rationality on the part of the one who is to be controlled 
would probably lead to erroneous conclusions; i.e., although budgets are 
to be set so as to optimize performance, it is not believed that department 
heads will maximize monetary gain in an incentive system or have the 
weakly-ordered preference field required for rationality. Hence the 
results of the experiments discussed here may be relevant, but ^ith 
serious limitations. 

3.5. Some Field Studies and a "Practical" Example 

The studies at the Hawthorne plant of the Western Electric Company 
are classic in the field of industrial psychology. ^ Among other things, 
''The experiments served an important purpose in calling attention to 
the fact that interpersonal relations and the character of the social situation 
can alter the effects of such [rest pauses and wages] specific incentives. "- 
The evidence which is perhaps most important here, however, is the 
indication that change per se appears to be a factor in the output increase 
in the selected group of Hawthorne workers. Furthermore, there is no 
indication that the workers reacted unfavorably either to the increase 
in their hourly effort expenditure or to the emphasis on productivity 
(in terms of the incentive system and unaccustomed frequency of output 
measurements). If the Hawthorne results could be extrapolated to 
nonproduction workers, then they would clearly imply that changes in 
social factors might be employed to mitigate the dissatisfaction resulting 
from the expenditure of additional effort to improve performance. 

A study whose findings are contrary to the usual assumption that 
people work better under permissive than under restrictive management 
is also of interest here. This study, which is nearly unique (among the 
studies reported here) in that it observed someone other than production 
workers, was performed by Wechsler, Kahane, and Tannenbaum (84). 

The subjects were two divisions of employees including plwsicists, 
engineers, scientific aides, and supporting clerical personnel, designated 



iFor a summary, see (82), pp. 181-193, 214-215. 
'-Ibid., p. 193. 



Available Empirical Information 57 

as Division A and Division B. The head of Division A, containing 28 
people, was a young brilUant scientist whose leadership was restrictive; 
Division B, containing 38 people, was headed by a less ambitious older 
man who exercised permissive leadership. Employee questionnaires 
indicated a considerably higher morale level in Division B. Although 
about 57 per cent of the employees of both divisions considered the 
productivity of their work group (a subgroup of a division) "high" or 
'Very high" and only 3.6 per cent of Division A and 5.2 per cent of 
Division B considered the productivity ''low" or very "low." The corre- 
sponding estimates of over-all Division B productivity by its employees 
were 55.2 per cent and 7.9 per cent, but for Division A 28.6 per cent and 
18.6 per cent. The employees of the permissive Division B considered 
the productivity of the laboratory (consisting of several divisions) above 
average in 26.3 per cent of the cases, below average in 13.2 per cent; 
whereas the corresponding percentages in the restrictive Division A were 
7.1 per cent and 32.1 per cent, respectively. Interviews were held 
with five superiors and two staff members who were familiar with the 
objectives and performance of the two divisions. The superiors rated 
Department A, in spite of its restrictive leadership, higher than Depart- 
ment B and higher than their own estimate of their performance. 

It appeared that the director of Division A set objectives for his 
division which were more or less identical with those of his superiors, 
and in spite of low morale and low job satisfaction was able to "lead" 
his group into fulfillment of these goals. However, the director of 
Division B "utilized the services of a high morale group and of satisfied 
people in the performance of tasks which his superiors did not consider 
of highest importance to the laboratory." ^ 

The study did not make clear precisely how much original research 
was conducted in the two groups or whether the restrictive leadership 
was equally binding on all members of Division A. It is also possible 
that more original work was produced in Division B while the objectives 
of the superiors may have been more short-sighted, reflecting an interest 
in problems of temporal rather than lasting interest. 

Despite the ambiguities in this study, however, and despite the 
difficulty which surrounds the assessment of a proxy objective (and 
output) for a laboratory, it is interesting to observe that the low morale 
group was the high production group in this study. ^ There exists a strong 



K84), p. 6. 

^Confirmed also by University of Michigan studies of worker morale and produc- 
tivity. 



58 Available Empirical Information 

indication, furthermore, that the low job satisfaction might be attributed 
to stress caused by a higher level of aspiration in Division A, since it is 
reasonable to assume that the estimates of group productivity were set 
with reference to a "norm" which would constitute a level of aspiration 
for the division. 

A practical example will be introduced to provide some evidence for 
the proposition that, whereas economics assumes an optimum production 
function, this is not likely to be the case in practice. Statistics compiled 
by the Lincoln Electric Company on their operations, for example, 
indicate a consistent reduction of cost and direct labor per unit output 
over time, both in absolute terms and relative to other producers of the 
same product.^ These data cannot be explained entirely in terms of 
economies or diseconomies of scale (since the firm has been both small 
and large relative to the other firms in the industry), nor entirely in terms 
of technological change (since either new technology or at least the 
opportunity to acquire it would appear to be shared by the industry as 
a whole). Alternate hypotheses might be (1) that the production 
function, relative to a given technology, may depend on factors (e.g., 
psychological) other than those usually considered in economics and 'or 
(2) that some or all firms in an industry do not operate at an optimum. 

In the case of Lincoln Electric, many incentive techniques have 
been used which would not ordinarily motivate ''economic man;" e.g., 
providing the workers with a sense of participation in management of 
the firm. In addition, an assumption which pervades Lincoln's approach 
(and is not altogether divorced from some of the assumptions which are 
made in the previous chapter) is the omnipresent possibility of improve- 
ment which may imply the existence of an optimum, but is incompatible 
with an assumption of current operation at an optimum. 

The foregoing would serve to provide some validation to the mvesti- 
gation of control mechanisms as devices which influence the approach to 
optimality, although the evidence provided by such "practical examples" 
is necessarily vague and inconclusive. Further evidence for the non- 
optimal behavior of business firms is available from the study of C3'ert, 
Dill, and March (23). 

A convincing theoretical treatment which concentrates on the 



^See (51) or (50). It should be noted that these data are company data and subject 
to obvious bias as such. However, the fact that Lincoln has lowered its price from 
$0.17 to $0.05 per pound for electrode and has, in the process, become the largest 
producer of that commodity, offers some supporting evidence for the above comments. 
Furthermore, these data have not, to my knowledge, been challenged by competing 
firms. 



Available Empirical Information 59 

approach to optimality, rather than the a priori assumption of optimah- 
ty, is presented by Charnes and Cooper (11). The approach they use 
embodies a more structured definition of the cost structure than is used 
here, and hence is of great interest in implementing the more exphcit 
statements of the psychological factors in this paper. 

3.6. Conclusion 

The studies presented in this chapter provide a cross section of the 
evidence available about human behavior which might be applicable to 
an individual in a budget-controlled activity. More, of course, could be 
presented, but it is hoped that this brief survey will at least provide 
some background for the analyses and experiments which will be dealt 
with in the following chapters. 

To summarize, there is evidence that individuals form either indi- 
vidual goals or estimates of their performance (or perhaps both). On the 
other hand, it is not precisely clear which of these is being formed at any 
given instant. These aspirations (or expectations) tend to be decreased 
following a failure in a previous trial in the same task, or increased by a 
success in that task. The aspirations are affected by external reference 
points other than performance, but this effect tends to decrease as 
experience with the task increases. (However, the effects of rewards 
have not been clearly determined.) The aspiration level is subject to 
change with success or failure on related tasks, but it is not clear whether 
the effects can be explained by stimulus generalization or by similarities 
of need. 

Animal studies of motivation are a potential source of information, 
but the problems of inference relating them to human behavior are not 
solved even in general, so that use of information gleaned from these 
studies is of dubious value for the subject with which this thesis deals. 

Experiments in utility maximization have used, as a basic premise, 
the ''rationality" of man. On the one hand, these studies are by and large 
oriented only to an individual's tastes and performance. On the other 
hand, there is no universal agreement on the basic postulates. Further- 
more, these experiments have been conducted with amounts of money 
whose expected values are, as Dreze notes, in danger of being regarded 
as trivial by the subjects. This further attenuates the results secured 
(since they become even more difficult of extrapolation to an actual 
situation) and, hence, tends to reduce the reliability of conclusions that 
might otherwise be drawn. 

Field studies have shown conflicting results, and only isolated 



60 Available Empirical Information 

examples of studies which indicate the possibihty of introducing an 
adequate control scheme appear. A further weakness (from the stand- 
point of this study) is that almost all of the more substantial studies 
in this area have been directed primarily towards the behavior (motives, 
etc.) of production workers, as distinct from budgeting or budgeted 
management. 

Business experience, though voluminous, tends to be so loosely 
phrased (or reported) and to contain such a mixture of complex and un- 
resolved factors, that little can be gained, at this time, by a recitation or 
analysis of this experience. It therefore has seemed best to confine the 
presentation here to a single instance where the management has at least 
been more articulate than most. It is interesting — although, of course, 
not decisive — that this company (the Lincoln Electric Company) has 
issued its series of pronouncements in a form which is not wholly in- 
compatible with the theory covered in the preceding chapter. 



, 



CHAPTER 4 



An Experiment 



4.1. Introduction 

The brief survey of received evidence, analyses, and hypothesis 
(i.e., the survey just concluded) does not reveal any body of material 
which is sufficiently pointed either to validate or even to give satisfactory 
guidance for a theory of budgetary control of the kind which is of interest 
here. With this in view, an experiment was designed to see what could 
be uncovered by the laboratory techniques of experimental psychology 
when these are combined with the principles and tools of modern 
statistical inference, as exhibited by the theory of experimental design, 
and the tools provided by the analysis of variance, etc. 

In the experiment which will now be reported, a major objective was 
to investigate relations that might exist between individual performance 
and aspiration levels and the relations that might also exist between 
these variables and the kind of "external"^ goals which are represented 
by a budget of the kind commonly employed in management practice. 

From one standpoint this experiment may be regarded as an attempt 
to repair existing gaps in the extant literature. In particular, it is 
designed to help remedy a situation where no satisfactory and explicit 
attempt has been made to ascertain the effects of varying aspiration level 
upon performance, and also to supply some modicum of evidence on the 
effects of explicit goals (e.g., those emanating from an experimenter) 



^I.e., goals which are at least partly external in the case of any single individual 
where the term "external" allows the possibility of a group or other norm to which 
an individual may willingly (consciously or unconsciously) conform. 

61 



62 An Experiment 

as they bear on either (or both) of the other two variables. From another 
standpoint the results of this experiment may (it is hoped) be regarded as 
a beginning towards the accumulation of a systematic body of evidence 
upon which an improved understanding of the problems of budgetary 
control may be built. 

It is to be emphasized that the results of this one experiment do not 
warrant any firm conclusions on the problems of budgetary practice 
as they are found in actual management systems. Nevertheless, a 
beginning, if it is to be made somewhere, might as well start with the 
kinds of factors — external goals, aspirations, and performance — that 
are included in this experiment. In the present stage of knowledge, 
the advantages of a laboratory test appear to outweigh anything that 
might be gleaned from alternate approaches; e.g., a broad survey of 
existing industrial practice or intense studies of a few selected cases. ^ 

The budgets used in the experiment are of a kind sometimes referred 
to as "performance budgets. "^ By this it is meant that the measures, 
hence the controls, are gaged with respect to physical quantities only. 
"Cost budgets" — i.e., budgets cast in terms of dollar amounts — are 
not covered. 

It may be recalled at this point that the models and analyses of Chap- 
ter 2 were couched in terms of "cost behavior" rather than "phj^sical 
performance" as such. Formally, the difference is easily attended to. 
For instance, a substitution of p (performance) for c (cost) in equations 
(2.5.1) and (2.5.2) accompanied by the use of (a — p) rather than (c — a) 
as a measure of stress associated with aspiration level discrepancies 
provides a restatement of the requisite kind. Correspondingly, the 
technological conditions would also be restated by replacing (c — c) with 
{p — p) in the exponential. But of course, matters of formal equivalence 
do not necessarily establish or resolve empirical questions which involve 
actual (psychological) behavior responses. It is possible, for example. 



iln any event such surveys and case studies have been made by others, and reports 
of results are available in the standard literature. 

2This should not be confused with this same term as it appears in The Hoover 
Commission Report on Organization of the Executive Branch of the Government (36). 
Cf., e.g., p. 36, Recommendation No. 1, which proposes that, "The whole budgetary 
concept of the Federal Government should be refashioned by the adoption of a budget 
based upon functions, activities and projects: this we designate as a 'performance 
budget'." Also, further on, "Under performance budgeting, attention is centered on 
the function or activity — on the accomplishment of the purpose — instead of on 
lists of employees or authorizations of purchases ... It places both accomplishment 
and cost in a clear light before the Congress and the public." 



An Experiment 63 

that budgets cast in terms of cost might have effects which differ from 
those couched in terms of physical (performance) measures only. It is 
also possible that budgets stated in both cost and physical terms may 
have still different consequences, since logical and psychological equiv- 
alences are not necessarily isomorphic.^ 

Bearing these kinds of considerations and qualifications in mind, the 
gist of the experiment actually undertaken will now be described. 

4.2. The Experimental Task 

Each subject took a series of 6 tests, each containing 15 water-jar 
problems of the type used by Luchins (52). ^ In each of these problems 
a subject is told that he has 3 empty jars of different capacities (expressed 
in integers). He must, without approximating, fill the jars from a tap 
and then empty them into a sink or into one another in such a way that 
he obtains a required (integral) number of quarts as the total amount of 
water in the jars at the end. A simple example will be more enlightening 
than a further explanation of the rules of the game. A 2-jar problem 
might be as follows: "You have a three-quart jar (A) and a two-quart jar 
(B), and you are required to obtain one quart." An appropriate solution 
would be: 'Till A, fill 5 from A, empty j5," leaving the required one 
quart in A. A second solution would be "Fill 5, fill A from B, fdlB, fill A 
from B, empty A," leaving the required one quart iuB. Either sequence 
of steps would be considered correct. It will be noted, of course, that 
either of these solutions can be expressed algebraically (viz. A — B or 
2B — A), but the problems were "sugar-coated" in order to avoid 
possible antimathematical "blocs" associated with presenting them as 
problems in algebra. 

The 6 tests, as constructed, were intended to be "equally" difficult. 
But, as the subsequent statistical analysis revealed, this objective was 



^Thus, it is to be noted that the formal (mathematical) transformation introduced 
does not really come to grips with issues often discussed in the budgetary literature 
on such matters as to whether overhead and fixed elements of cost should be included 
in the budget or to whether each supervisor should be budgeted (hence held account- 
able) only for variable items of expense that he can directly control (rather than 
merely indirectly affect). Furthermore, the experiment does not really deal with 
"group endeavors," nor does it deal with issues such as "cost circulation" and related 
allocation issues. Some of these topics will be discussed later in this paper but without 
the benefit of experimental evidence — which it is hoped will be generated at a later 
time. 

2A sample test and instructions given to the subjects are shown in Appendix 4A, 
pp. 96-102, 



64 An Experiment 

not wholly achieved. A general description of the choice of the tasks is 
as follows:^ The capacities of the jars were chosen at random. The 
solutions (i.e., formulae expressed in terms of jar capacities which ^-ield 
the required number of quarts) were chosen at random from subsets of 
possible solutions which were constant from test to test. In order to 
provide for a desired rough ordering of problems (in terms of difficulty), 
the subset from which problem 1 was chosen contained very simple 
solutions; whereas the subset for each subsequent problem was deter- 
mined by eliminating some of the simpler solutions contained in the 
subset for the previous problem, while adding an equivalent number of 
more difficult solutions. 

The selection process was intended to provide a gradual increase in 
the difficulty of the problems on any one test as the subject worked from 
beginning to end. As a result of various peculiarities in the problems 
(explained in detail in Appendix 4A), it would have been difficult, if not 
impossible, to determine an a priori measure of difficulty sufficient to 
provide a precise ordering, even if the overlapping of solution subsets 
were to have been eliminated. Examination of the experimental data 
indicated that, among other things, the ordering depended upon the 
particular skills of a subject. Uniqueness of the solutions could not be 
guaranteed nor, as will become evident in the following remarks, was it 
considered advisable to eliminate problems having a multiplicity of 
solutions. The measure of performance used was the number of problems 
correctly solved (independent of difficulty) in the seven-minute trial 
period allowed for each test. 

With this much of the experimental background at hand and with 
some of the attendant difficulties now set forth, it may be helpful to 
examine these issues from the standpoint of similar problems as they 
might be expected to occur in situ in industrial practice. A simple 
illustration would be one involving a department head, a foreman, or 
other such individual in the echelons of "budgeted management." Let 
it be supposed that this individual is given a set of problems (or tasks) 
which involve some degree of intellectual application combined with 
judgment in varying degree; and let it be further supposed that (a) a set 
of standards is imposed and that these vary by some presumed order of 
difficulty, although (b) not all of the problems have a unique solution. 

Note that the goals or other criteria incorporated in the varying 



^A description of the selection process utilized in the determination of the task, 
sufficiently detailed to allow reproduction of the tasks used here (subject to random 
deviation), appears in Appendix 4A, pp. 93-95. 



An Experiment 65 

standards may not correspond to "real" orders of difficulty and, a 
fortiori, may not conform to the order of difficulty which a particular 
manager might either find apparent or else experience in actuality. 
Moreover, neither the standards nor the actual performance may meet 
the optimizing requirements of the economic theory of production — 
in which nonuniqueness is resolved by the assumption of optimization 
per se.i 

As already noted, no claim is made that the results of this study are 
immediately ready for extrapolation and application to industrial 
practice. Nevertheless, the motivation of the study is such that the 
relevance of the theories, designs, etc. for general classes of problems, as 
they might be encountered in management practice, was constantly borne 
in mind, and the resulting experiments were judged, so far as possible, by 
reference to this kind of possibility. From this standpoint, and within 
the limits already noted, it is therefore interesting to note that the 
features of task selection which caused difficulty from a purely scientific 
standpoint are of the same kind that might be expected to prevail, at 
least under certain circumstances, in practice. 

In the experimental setting (lacking an explicit formulation of cost or 
production as a function of problems solved), the measure of performance 
chosen was the number of problems per se. The time limit is in accord 
with the hypothesis implied in the model of Cooper and Charnes (11), 



iThe imposition of optimality conditions, as found in economic theory, provides 
a convenient way of reaching uniqueness. But actually this resolution is assumed, 
rather than proved (or empirically validated), in economic theory. Cf., e.g., A. 
Charnes and W. W. Cooper (11), or Sune Carlson (5), pp. 14-15: ". . . If we want 
the production function to give only one value for the output from a given service 
combination, the function must be so defined [sic] that it expresses the maximum 
product obtainable from the combination at the existing state of technical knowledge. 
Therefore, the purely technical [sic] maximization problem may be said to be solved 
by the very definition of our production function." Presumably Carlson, as a typical 
economist, means that someone else (e.g., engineers) solve this problem so that 
learning and other such adjustments are absent. But as R. Dorfman, P. A. Samuel- 
son, and R. A. Solow (26) somewhat humorously note in Linear Programming and 
Economic Analysis, p. 131: 

The production function is a description of the technological conditions of 
production, and the economist takes no direct responsibility for ascertaining it. 
Instead he regards it as falling within the purview of the engineer. But there 
seems to be a misunderstanding somewhere because the technologists do not take 
responsibihty for production functions either. They regard the production 
function as an economist's concept, and, as a matter of history, nearly all the 
production functions that have been derived are the work of economists rather 
than engineers. 



66 An Experiment 

where it is assumed, when learning and adjustment are allowed for, that 
a supervisor's ability to approach optimality in a particular phase of 
operation is limited by the amount of time he can devote to that phase. 
Likewise, the problems which occur in an industrial setting do not 
necessarily have a unique solution, and it is reasonable to assume that 
frequently problems are solved just as effectively (except for the addi- 
tional time involved) by a roundabout method even though a simpler 
solution may exist. 

From a statistical standpoint, the rough ordering introduced was 
utilized as a means for avoiding the possibility of extremely large 
differences in scores arising between subjects who proceed from easy to 
''more difficult" problems, in contrast to those subjects who might either 
proceed on an opposite course or else "plod straight through" the 
problems in the order given in a particular test. Such differences might 
easily produce bi-modal, tri-modal, or (in general) multi-modal distri- 
butions and thereby (a) obscure the main effects or (b) require recourse 
to complicated statistical analyses. 

The reasons for introducing problems of varying difficulty may also 
need to be explained. First, despite the ranking difficulties alread}^ 
discussed, it is still harder to design a battery of "equally difficult" 
problems. A possible exception is, of course, the case where all problems 
in a test are exactly the same.^ If this were an experiment in, say, rote 
learning, such an approach might have been adopted. Not only was this 
not the case, there were also difficulties that might be anticipated in view 
of the choice of experimental subjects (university students) who might 
react to the resulting ennui with complete diminution of effort. Second, 
there was some interest in ascertaining the effects on performance vrith 
variations in task difficulties. Inter alia it was desired to ascertain 
whether consistency of performance might be achieved via a learnmg 
route under which experience carried over from simpler problems was 
applied (later) to more complex ones.^ Third, a range of available 
problems introduced a safety factor in the design which could be utilized 
if analyses — carried on pari passu with the tests — showed anythmg 
to be going seriously awry. For instance, a preliminary inventor}^ of 
more difficult problems would be available in case the motivation (or 



iBut note that issues still arise in connection with repeated solutions of the same 
problem; e.g., discouragement associated with initial failure as well as the "jumps'' 
which are likely to occur after a first success is achieved. 

2It should be noted that, although the tests were of (approximate!}') equal diffi- 
culty, the "budgeted learning" was such that the subject was expected to do more 
(and hence more difficult) problems as he proceeded through the series of trials. 



An Experiment 67 

other) analyses showed any lack of interest or boredom on the part of a 
significant number of the subjects already examined.^ Finally, the use 
of a range of more and less difficult problems made it possible to analyze 
these effects per se — and these effects are not uninteresting for the 
problems of practice — and also avoided the need for undertaking 
unnecessary extrapolations in this direction on the basis of a more limited 
range of evidence. 

With this much of the reasoning now explained, it is desirable to turn 
to a discussion of other aspects of the experiment; namely, a problem 
involving not merely ''intellectual" (e.g., learning) carryover, but also 
carryover of problems existing in the industrial setting. It was, in fact, 
found to be impossible to ascertain, even approximately (in the exper- 
imental setup), the effect of an inventory of unsolved problems which 
''carries over" from one period (or trial) to the next. Nor was it deemed 
desirable to complicate matters by utilizing an approach which intro- 
duces certain "crucial problems," which must be correctly solved as a 
precondition for receiving recognition for any performance at all. Such 
an approach, of course, has a certain appeal from the standpoint of its 
potential bearing on the problems of actual budgetary applications in 
management so that its omission is justified only by expediency and 
immediate practicality. It's omission from the list of experimental 
factors mentioned here may therefore be regarded as one more qualifi- 
cation relative to the problems of actual budgetary practice. 

4.3. The Reward Structure and Budget 

A "budget" 2 was set for each subject for each test. Before the start 
of the trials, each subject was given $3.00. If the subject "made his 
budget" (solved the budgeted number of problems or more) he was given 
an additional dollar. If he solved fewer problems than the budgeted 
number, he was required to pay the experimenter $1.00 from his pre- 



^The problems which arise in the construction of equally difficult (but different) 
tasks for subjects in a situation in which learning is a factor are not trivial. Chapman, 
Kennedy, Newell, and Biel (8) found that a group of college students "learned its way 
right out of the experiment," enabling them to perform adequately while engaging 
in nontask activities while a trial was in progress. Based on this experience, suc- 
ceeding trials in their experiment were designed to reflect the hypothesis that task 
difficulty was not determined by the number of problems per se but by the number 
of problems relative to the group's (learned) problem-solving capacity of the moment. 

2For fear of possible misunderstanding, the term used with the subjects was 
"goal." Cf. Appendix 4A. 



68 An Experiment 

viously accumulated sum. Thus both positive and negative rewards 
were involved, and a potential difference of $2.00 in asset position was 
at stake in each single test of seven-minute duration. ^ 

As already noted, each subject was given $3.00 at the start of the 
experiment. In order to avoid automatic failure via exhaustion of 
resources on the fifth or sixth test, each subject was allowed to make a 
dollar on either of these. In any event, no subject w^as required to deplete 
his own funds beyond the loss of the initial S3. 00 plus accumulated 
earnings. In sum, any subject could theoretically leave the experiment 
having earned as much as $9.00, or as little as nothing. ^ 

The subjects were divided into four groups of equal size and labelled 

A, B, C, and D according to the kind of budgetary situation for which 
they were to be tested. Group A represents what will be called an 
''implicit-budget" group. This group was never told what their budgeted 
amount on any test was to be. In contrast, Groups B, C, and D will be 
called the "explicit-budget" groups. Where explicit budgets were 
involved, subjects were informed of their budgets before each test 
trial. 

In addition to the explicit-implicit classification, the budgets applica- 
ble to the four groups were further subdivided into a ''low," "medium," 
and a "high budget." On the first test for all groups (implicit as well as 
explicit) the budget was set at 5 problems (correctly solved). In Group 

B, the "low budget" group, the budgets for the subsequent tests were 
obtained by adding alternately 1 and to the previous test score. For 
Group C, the "medium budget" group, the "adjustment" was alternate^ 
2 and 1; and for Group D, the "high budget" group, 3 and 2. Group A 
(implicit budgets) was further subdivided into three groups, each third 
having a budget determined in the same way as one of the "explicit 
budget" groups. 



iThe experiments of Mosteller and Nogee (63) used amounts whose expected value 
was at most $0.10. Davidson, Suppes, and Siegel (24) used amounts up to SO. 50, but 
only rarely. Edwards (29) used prizes which ran as high as $6.00. (It is proposed in 
a subsequent experiment to ascertain the effects of variation in monetary rewards.) 

^Actually, if the subject's stock of capital was less than $3.00 at the beginning of 
the sixth test, the grading (done away from view) was arranged so that the subject 
automatically "made the budget" regardless of his score or, if a subject's stock was 
$8.00, he automatically failed to make the budget on the last test. (The first few 
subjects, depending upon their stock of capital, were given an easy, normal, or 
difficult seventh test on which the reward or penalty was $2.00, but this increased 
the time considerably without supplying much additional information. In view of 
this a switch was made to the above procedure, which, as far as could be ascertained, 
worked satisfactorily.) 



An Experiment 69 

An issue which is often of interest, at least as far as the literature 
of budgeting and standard costing is concerned, deals with presumed 
advantages and disadvantages of optimal (or ideal) standards versus 
'Hight-but-attainable standards" and budgets which are "too loose" or 
''too tight. "^ These statements are usually couched in loose and very 
general terms which, apparently, are intended to apply over a great 
variety of situations as well as to persons of differing socio-psychological 
makeups and needs. ^ 

It is of interest in this connection to note that the ''low budget" of 
the experiments is the only one that may be said to correspond to the 
often-cited principle of "attainable but not too loose. "^ In contrast, 
the principle of budgetary adjustment utilized would require, for the 
medium budget groups, successive attainments in the amount of 5, 7, 
8, 10, 11, and 13 (or better) problems correctly solved as the trials 
proceeded.^ These figures assume an additional constraint — viz., that 
a subject running through the indicated sequence of trials could do, at 
most, one additional problem on 2 tests or 2 problems on one test. (This 
allows for the rule for adjusting the succeeding budgets, relative to past 
performance, that was employed.) Finally, the "high budget" was 
mathematically incapable of attainment over all trials so that it lies at 
the opposite end of the spectrum. 

A word also needs to be said about the implicit-budget category which 
was also (it will be recalled) subdivided into low, medium, and high for 
test purposes. In part, this group served the purpose of a statistical 



^Cf., e.g., Walter B. McFarland (61) for a succinct summary. See also, R. M. 
Trueblood and W. W. Cooper in their article "Cost Accounting" in the Encyclopxdia 
Britannica (79). 

2See, e.g., Stanley B. Henrici (35), Chapter III ff. It is to be noted that Henrici, 
in common with many writers, is inclined to draw overly fine distinctions between 
"budgeted" and "standard" costs since, as he himself admits on p. 29, "As a matter 
of fact, both [i.e., budgeted costs and standard costs] may be used coordinately in the 
same enterprise. The difference [if any] lies in the use to which they are 
put . . .." 

Wide, e.g., McFarland, op. cit., p. 154: ". . . in the related field of time study . . . 
it seems generally agreed that a standard rate should be attainable by a majority of 
the workers." See also Henrici, op. cit., p. 30: ". . . the only arguments against 'ideal' 
(i.e., optimum) standards is that they are difficult to sell ... [It is important] that the 
supervisors will know that they have been set a task possible of achievement." 

^Actually only one subject in the B classification attained the budget on every test, 
although 4 subjects in other groups had scores sufficiently high so that (assuming 
away budgetary- and aspiration-level effects) their performance would have clocked 
off the required minimal amounts in every trial of the B group. 



70 An Experiment 

control in order to handle confounding which might otherwise occur. 
But it is also of interest in its own right. The practice of implicit budgets 
is probably still widespread in industrial practice. In particular, even 
firms with developed budgetary practices do not necessarily have every 
item, every supervisor, and every operation necessarily encompassed in 
the formal budgeting procedures.^ In fact, it is common, even in the 
literature, to distinguish between ''complete" and "incomplete" budgets 
to accommodate these kinds of cases. ^ 

It cannot be safely assumed, however, that merely because a system- 
atized and formalized procedure is absent, supervisors not covered by 
an explicit budget do not then believe that others (e.g., their superiors) 
hold no expectations concerning their behavior. And, after all, the 
essence of the budgetary control procedures is formulated around such 
"external" expectations which become crystallized as a stated goal when 
formalized in either a budget, a physical or cost standard, etc. 

To state the matter differently, it would have been possible to effect 
the indicated statistical resolution by introducing a "no budget" group 
for control purposes. This was not done, however, for the reasons already 
stated: It is difficult to conceive of a managed situation under which 
some one or more individuals (who are subordinate to others) would be 
given a completely free hand without any notion that others were at least 
informally forming goals or standards by which to judge their perform- 
ance. It therefore seems logical to turn in this direction and to introduce 
the additional replicates involved — i.e., a low, medium, and high 
implicit budget — to satisfy the statistical requirements when this has 
been done.^ It was then felt (or rather hypothesized) that the groups 
would attempt to infer from the resulting penalties or rewards what kinds 
of (implicit) budgetary levels were involved. 

It is important now to detail the kind of information supplied to the 
subjects with respect to the budget-setting and accounting procedures. 



iCf. R. M. Trueblood and W. W. Cooper, loc. cit. 

2Thu8, E. L. Kohler (44) observes, on p. 67, under the term budget: ". . . Complete 
budgets covering all phases of an enterprise as well as incomplete budgets covering 
only certain phases are also met with in practice." 

^If such replications are not introduced, it becomes necessary to have recourse to 
rather involved statistical formulas, many of which have to be worked up de nouveau 
since, apart from so-called missing lot-plot analyses (generally of an agricultural 
variety), the literature of statistics does not provide readily available formulas, but 
only general guides to the kinds of adjustments that would be needed — including 
adjustments for unequal subgroups, etc. 



An Experiment 71 

Here again it was desired to conform as closely as possible to situations 
that might be encountered in practice. In particular, it was desired to 
reproduce (in so far as this was possible) a situation in which the subjects 
know only general policies — often in a somewhat vague form — and 
have only a loose notion of their actual attainments as gleaned directly 
from their own experience. There is also a question of ''informal" as well 
as ''formal" policy to be dealt with. Thus an actual budgetary adjust- 
ment process may be utilized (as distinct from one which is formally 
stated) and the individual left to his own devices insofar as inferences 
about this process are concerned. 

In the tests as run, the subjects were informed verbally that the goals 
were set "according to your individual performance." Subjects were not 
informed of their scores and the tests themselves were designed so that 
the subjects would not be likely to know their exact numerical scores — 
i.e., number of problems successfully solved. A posteriori, most subjects 
did not work straight through and few counted the number they had done; 
furthermore, the probability of error was high so that even an actual 
count would not suffice to give a firm answer on the number of problems 
correctly solved.^ 

The vagueness of information supplied to the subjects combined with 
the alternating adjustment appeared to be quite effective in camouflaging 
the budget setting process in all but a few cases. The main purpose of 
not revealing scores, however, was to reduce the amount of information 
the subject received to compensate for the direct correspondence between 
budgeted number of problems solved and budget performance. 

Further discussion of this topic is now best delayed until the results 
of the statistical analyses can be presented. 

4.4. Aspiration Level Determination 

Attention will now be turned to the methods that were utilized for 
determining aspiration levels. 

It will be recalled that the interest of the study for this factor extends 
to possible relationships between budgets, aspiration levels, and per- 
formance. For this purpose each of the four budget groups — A, B, 



^The possibility of error would, of course, be expected to lead to a somewhat biased 
estimate of the score. Purely qualitative indications of this bias — viz., expressions 
of disappointment upon failure to receive reward — were received by the experi- 
menter. No attempt was made to ascertain the effects of this bias on aspiration 
and/or performance. 



72 An Experiment 

C, and D — were further subdivided into three groups, a, /?, and 7, 
according to the time precedence for securing aspiration levels and 
presenting budgetary information. In group a no aspiration level 
statements were called for. In Group ^ aspiration levels were stated in 
advance of any budgetary information, whereas in Group 7 exactly the 
reverse information process was employed so that this group had the 
requisite budgetary information at hand before stating its aspiration 
levels. 

A description of this part of the design may, perhaps, best commence 
with Group (3 — i.e., the group consisting of A/3, B/3, C/3, and D/3. Before 
each test, but after the rewards and penalties for the previous test had 
been distributed, each subject was asked ''How many problems do you 
hope to complete on test X?"^ To avoid, or at least to avoid reenforcing, 
a plausible (but untrue) inference that his aspiration level statement 
might be used in setting subsequent budgets, the following procedure was 
employed. Each ^ subject was instructed to turn the questionnaire face 
down before him as soon as he had answered the questions. He was told 
also that none of these questionnaires would be collected until all six 
tests had been completed. On the other hand, to eliminate the possibihty 
of ex post facto adjustments of stated aspirations by reference to actual 
performance, the subject was not allowed to change any statement on a 
turned-over questionnaire or even to refer to it when completing sub- 
sequent questionnaires. 2 

Only after ^ had stated their aspiration levels was the budgetary 
information given to them for the next test. And only after all tests had 
been completed — each test being collected at the end of each trial — 



^Certain additional questions, as exhibited in Appendix 4A, were also asked in a 
not wholly successful attempt to validate, or at least qualify, whether or not the 
received statement actually represented a "true" aspiration level. Also, it should 
probably be noted that rewards and penalties were simultaneously distributed among 
the subjects at the same time as the first piece of paper (questionnaire or budget "i was 
distributed, during the interim periods which obtained between tests. On this first 
piece of paper the qualitative score (success or failure) was also noted for each 
subject: a( + ) if he had attained the budget on his last trial and a( — ) if he had not. 

20f course, in actual budgetary practice such aspiration levels may have exactly 
the indicated adjustment effect via the negotiation (and signature) processes that are 
often associated with budget setting. On the other hand the context of this experi- 
ment indicated that such effects might be highly unpredictable and possibly swamp, 
via variance increases, other aspects of the study which were more immediately 
interesting. The larger number of trials or resulting complications in the design and 
analyses necessary to overcome this possibility were not deemed to be warranted 
in terms of the choices that necessarily had to be made within the study Umits. 



An Experiment 73 

were the questionnaires for this group collected.^ (These questionnaires 
had, of course, been pre-coded for purposes of collation in the subsequent 
statistical analysis.) 

For the 7 group — i.e., A7, B7, C7, and D7 — the rewards, penalties, 
qualitative scores, and budget information for the next trial were 
distributed at the same time as the questionnaire. Hence each 7 knew 
his budget before being called upon to state his aspirations. 

Finally, for Group a, no aspiration level statements were asked for. 
Here the subjects were given their rewards and penalties, the relevant 
budgetary information (as they fell into implicit or explicit budgetary 
groups) along with the tests for the next trial. This group could therefore 
immediately set to work, without being bothered to supply information 
on the intervening variable, ''aspiration level. "^ 

An attempt was made to find "minimum goals" for the subjects as 
well. 3 However, the question, "Do you believe that you should be 
penalized if you do not complete as many problems as you hope to?" 
was usually answered in the negative. Hence the answer to the third 
question was frequently nonexistent, and also some strange results (such 
as a minimum goal greater than the aspiration level) appeared. The 
questionnaire following the test gave indications that several subjects 
did not understand the difference between questions 1 and 3, so that 
the data were considered too unreliable for use. 

One further point should perhaps be noted with reference to budget 
group A in that it required some special handling and treatment. In 
order to differentiate between Groups A/3 and A7, the A7 subjects were 
asked first to guess what the budget would be and then, assuming this 
to be the actual goal, form their aspirations. The presence of a goal 



^Analyses of supplementary (motivational) questionnaires indicates that the 
procedure employed had some undesirable effects in that some of the subjects felt a 
sense of futility in forming and stating aspirations which they perceived as having 
no normative consequences. The results indicate, however, that these subjects 
underestimated the effects of the mere formation of these aspiration levels on their 
performance. 

2This may have given this group some advantage, not merely because of the 
(slight) saving in time and energy, but also because of the fact that it was not required 
to divert attention to matters other than the immediate task. On the other hand 
(even though the literature of budgeting is largely silent on this point), this kind of 
advantage may be associated with a lack of need for stating aspiration levels as well 
as other "attention diverting" procedures that are associated with normal budgetary 
practice. (In this connection see also the preceding footnote.) 

^Through the use of question 3 on questionnaires given to /3 and 7 subjects. See 
Exhibits 4A.3 — 4A.7, pp. 103-104. 



T4 An Experiment 

was thus made more evident in the A7 than the A/3 group duiing the 
aspiration level formation. Group A7 subjects received a piece of paper 
before each test which was merely a reminder of the existence of the 
budget. 

4.5. Experimental Procedure 

The subjects were tested in groups, ranging in size from four to 
seventeen. Since the aspiration level groupings required different types 
of paper distribution and instructions during the interim periods, it was 
necessary to run each group of subjects as all a, all (3, or all 7. However, 
all budget groups were represented in each group, the alphabetical 
ordering of the groups following some convenient path around the 
experimental room. Care was taken, however, to avoid bias in placing 
the subjects — e.g., the kind of bias that might be associated with placing 
all of the Group A members in the front row. 

The seating of the subjects, reading and explanation of the instruc- 
tions, the first aspiration level determination (in Group (3 and 7), and the 
first budget distribution consumed about twenty minutes. After each 
test (of seven minute duration) there was an interim scoring period which 
varied between seven and twelve minutes^ during which a discussion was 
conducted either on some topic the group was studying or the campus 
redevelopment program. This was designed to provide rest for the 
subjects as well as to avoid discussion of the experiment — as the amount 
of information dispensed would have been impossible to control from 
group to group. After the sixth test, a questionnaire designed to evaluate 
some general reactions of the subjects to the experiment was distributed. - 
The total running time for the experiment averaged two hours, varying 
about fifteen minutes in either direction. 



4,6. Structure of the Design and Subjects Utilized 

The number of factors to be considered and the number of repli- 
cations desired required careful consideration, at least from a statistical 
and resource requirement standpoint, since otherwise a forbiddingly 



^Depending upon the size of the group and the experience of the scorers. During 
the five days in which the experiments were run the efficiency of test scoring increased 
considerably. However, the aspiration level groupings were well scattered throughout 
the period so that no consistent bias was likely to have been introduced as a result. 

2A specimen is included in Appendix 4A, q.v. 



An Experiment 75 

complex and time consuming task would be confronted. In brief, the 
design involved 12 groupings (3 aspiration levels and 4 budget types) 
performed over 9 replicates giving 108 subjects, each involving 6 trials 
(or tests), for a total of 648 observations to be analyzed. Thus, in 
statistical jargon, the design used was a ''6 X 3 X 4 factorial design with 
9 replicates." In this context issues of convenience were sometimes 
favored over statistical niceties such as efficient designs and test power 
(relative to Type II error). ^ 

All 108 subjects were students at the Carnegie Institute of Tech- 
nology and all but 9 were studying management or administration. 
The range of subjects covered by year of college level attained ran 
the gamut from sophomore undergraduates to second-year graduate 
students. 2 In order to prevent large-scale information dissemination 
which might affect the performance of later relative to earlier subjects), 
two procedures were followed which necessitated running the subjects 
in large, homogeneous groups: (1) the tests were run over the shortest 
feasible (five-day) period, and (2) subjects who were most likely to have 
contact with one another were run simultaneously. This resulted in some 
confounding of the aspiration level groupings with subject classifications. 
However, this was in part controlled by having each set of replicates 
balanced (e.g., sophomore undergraduates offset by second year grad- 
uates) so that the effect of such heterogeneity would, in general, be 
reflected in an inflation of the remainder sum of squares and thereby 
compensate for a possible increase in the sum of squares due to the 
confounding of the aspiration level groups.^ 



^Actually, the question of Type II error in purely scientific work is not a wholly 
settled one in statistics. (See, for instance, R. A. Fisher's review (32) of A. Wald.) 
But, of course, such questions would, in general, be highly relevant for applied work 
and would naturally lead towards something like the sequential designs developed 
by H. Robbins and others (65) as an outgrowth of the principles underlying modern 
statistical decision theory as developed by A. Wald (83) and others. 

20f the students studying management, 12 were senior electrical engineering 
students in a management option curriculum, 42 were undergraduate students in 
Industrial Management, and 45 were graduate students in Industrial Administration. 
The remaining 9 consisted of 4 seniors in electrical engineering or physics and 5 
advanced graduate students in electrical or chemical engineering. 

^The classification of subjects in experimental groups is shown in Appendix 4A. 
An a priori measure of the group ability was assumed to be "aggregate years of study" 
— i.e., sophomore undergraduate ^ 2, junior = 3, etc., up to second year or more 
advanced graduate = 6. This quantity was limited to a range of 36 to 41 except for 
Group Aa which was low (32) and although not the lowest in performance, was lo\y 
compared with what might have been expected. 



IT 



76 



An Experiment 



4.7. Analysis of Results — Performances 

The criterion of effectiveness of a particular combination of aspiration 
level determination and budget level was considered, in the light of the 
reasoning of Section 4.2, to be the average number of problems solved by 
the group per trial. A tabulation of this variate, by groups, is given in 
Table 4.1.^ The significance of the group differences in performance was 
tested, as usual, under what, in statistical jargon, is called the ''linear 
hypothesis:" 2 



(4.7.1) 



where 



(4.7.2) 



Vijki = fJi -\- ti -\- aj -{-hk + {ta)ij + (th)ik 
+ {ab)jk + (tah)ijk + e^jki 

i= 1, ...,6; i= 1, ..,3; A: = 1, ...,4; 1= 1, 



9 



yijki = performance of the Ith. individual on the 2th test in the 
jth aspiration level group and the kth budget group 

ju = universe mean (average performance) 

ti = effect of the ith test on performance 

aj = effect of the yth aspiration level group on performance 

hk = effect of the ^th budget group on performance 



TABLE 4.1. AVERAGE NUMBER OF PROBLEMS SOLVED PER TEST BY 

EXPERIMENTAL GROUPS CLASSIFIED ACCORDING TO BUDGET TYPE 

AND ASPIRATION LEVEL DETERMINATION. 



\Aspiration 
^\Level 

Budget ^\ 


or 





7 


Average 

all 

Aspirations 


Budget 
Only 


Aspiration, 

then 

Budget 


Budget, 

then 

Aspiration 


A — Implicit 


4.56 


5.41 


5.60 


5.18 


B — Low 


4.09 


4.70 


4.56 


4.45 


C — Medium 


4.35 


5.45 


5.50 


5.10 


D — High 


5.13 


4.04 


5.85 


5.01 


Average 

all 
Budgets 


4.53 


4.90 


5.39 


4.94 



Note: Each of the subgroup averages represents the performance of 9 subjects in 
tests = 54 observations. 



^A more complete tabulation, showing total group scores on each test is given 
in Appendix 4B. 

2This should not be confused with the assumption (not made) that the relations 
to be tested are linear. See H. B. Mann (55), pp. 23, 24. 



An Experiment 



77 



and where (ta)ij is the effect on performance of the interaction of the ith 
test and the jth aspiration level grouping which cannot be attributed 
to the sum of the separate (main) effects, etc. The term eijki is a random 
error term, and in accord with standard statistical assumptions the errors 
are assumed to have zero expected value, and to be uncorrelated and 
homoscedastic. All other terms, save jjl, have zero first moments, and 
E{ai) = al, so that they, too, are homoscedastic.^ 

In general, it is assumed that the (mathematical and statistical) parts 
of the general Markoff theorem are satisfied^ and an analysis of variance, 
using the F-test may be employed. The results are as summarized in 
Table 4.2. The data indicate that a subject's performance is significantly 
affected by his budget, the way in which he determines his aspiration 

TABLE 4.2. ANALYSIS OF VARIANCE — PERFORMANCE 



Due to 


Sum of Squares 


Degrees 

of 
Freedom 


Mean Square 


Variance 
Ratio (F) 


Main Effects: 
Tests (0 

Aspiration 
Levels (a) 

Budgets (b) 


1557.500 

77.121 
53.315 


5 

2 
3 


311.500 

38.561 
17.772 


59.706t 

7.39lt 
3.406* 


Interactions (2-way): 
Tests X 
Aspirations (to) 

Tests X 
Budgets (tb) 

Aspirations X 
Budgets (ab) 


22.324 

21.728 

100.323 


10 

15 

6 


2.232 

1.449 

16.721 


.428 

.278 

3.205t 


Interactions (3-way): 
Tests X 
Aspirations X 
Budgets (tab) 


63.856 


30 


2.129 


.408 


Error 
(Remainder) 


3005.111 


576 


5.217 




Total 


4901.278 


647 







tDenotes highly significant (1 per cent level) 
*Denotes significant (5 per cent level) 



^See, for example, Kempthorn (41), Chapter 6; or Davies (25), Chapter 8. 
2See Kempthorn, op. cit., Chapter V ff. 



78 An Experiment 

level (e.g., before or after receiving budgetary information), and the 
combination of the two. There was a highly significant test effect but 
the magnitude of the differences between tests did not appear to be 
affected by the groupings — as indicated by the nonsignificance of the 
(ta) and (tb) interactions. 

Thus, it is possible to attach significance to the ordering of the 
performance by budget and aspiration level group. For notation 
purposes, let "X > Y" indicate that the subjects of Group X performed 
better (on the average) than the subjects of Group Y. Then, referring 
to Table 4.1, the relations 

(4.7.3a) A > C> D > B 

and 

(4.7.3b) 7 > ,8 > « 

are obtained. Thus the implicit budget group performed best, the 
medium budget group next, then the high budget group, and finalh' the 
low budget group. 

The group that determined its aspirations with knowledge of the 
budgeted goal (which was, on the average, higher than performance on 
all but the sixth test) performed better than the group without this 
knowledge and still better than the group w^hich did not expHcitly 
formulate its aspirations. 

Although the differences between the aspiration level groups taken 
in pairs are significant at the 2 per cent level or better and the differences 
between the low budget (B) and the other budget groups are significant 
at the .2 per cent level or better, the differences between A, C, and 
D taken in pairs are not particularly reliable.^ Thus the ordering, 
A > C > D, can be inferred from the sample as the most probable 
ordering, but it is impossible to reject the null hypothesis, i.e., A = C = 
D. 

Referring again to Table 4.1, the extremes of the table are D7 and 
D/3. This difference is estimated to be in the neighborhood of 6 standard 
deviations of the distribution of the sample mean and accounts in large 
measure for the superiority of 7 performance over /3 performance. 

An hypothesis which would serve to explain this difference is as 



^Assuming the sample means to be normally distributed and using the mean square 
error as the estimate of the population variance. The number of degrees of freedom 
for budgets (322) and aspiration levels (430) is sufficiently large to justify the as- 
sumption of normality. The probability of significant diflference is shown in more 
detail in Appendix 4B, p. 108. 



An Experiment 79 

follows : the D7 group formed its aspirations with the knowledge of the 
"high" management budget and hence tended somewhat to accept it as 
its own; the D/S group formed its aspirations prior to receiving its 
budgets in the light of previous performance, and when the high budget 
arrived mentally (i.e., psychologically) rejected it. 

Internal support for this hypothesis is available from analyzing the 
data on the relationship of the budget to performance. Although the 
budgets for the D/5 group were considerably lower than those for the D7 
group (since they depended upon prior performance), the D(3 subjects 
as a group attained the budget only 9 times during the entire test (or an 
average of 1 attainment per subject), while the D7 subjects attained the 
budget 19 times (or an average of 2.1 attainments per subject). 

An extension of this hypothesis to explain the superiority of BjS over 
B7 is, however, more dubious. The hypothesis would imply that the Bj3 
subjects, having previously stated their aspiration levels, would reject 
the budget as being too low and would strive for higher scores. The B(3 
subjects as a group attained the budget more often (3.7 attainments per 
subject) than the B7 subjects (3.2 attainments per subject), which would 
imply, if anything, that the B/3 subjects were influenced by the budget 
to a greater extent. On the other hand, the BjS group performances 
exceeded both their budgets and their aspiration levels by far more than 
did those of the B7 group, so that the situation is not at all clear. The 
budget attainments of 0(3 and C7 are almost identical (3.1 and 3.2 
attainments per subject respectively), as were their performances, so 
that little can be gained from further comparisons of this type.^ 

A note is in order regarding the overall performance of the subjects. 
Performance, p, can be expressed as a linear function of the number of 
the test, t, as follows: 

(4.7.4) pt = 2.37 + .732 t where ^ = 1, ..., 6. 

(±.0571) 

However, a component analysis of variance, as set forth in Appendix 4B, 
indicates that the quadratic and cubic components are also significant, 
so that the relation : 



(4.7.5) Pt = -0.65 + 5.71 t - 1.98 P + .212 t' 
or _ _ 

Pt = 4.22 - .338 {t- t) + .248 (t - ty 

+ .212 {t - ty 



1, ..., 6; t = 3.5 



lit should also be reported, for the sake of completeness, that A/3 subjects averaged 
2.6 attainments per subject, group A7 2 attainments per subject. For groups Aa, 
Bar, Ca, and Da, the figures are 2.1, 2.8, 2.7, and 2.6 attainments per subject, respec- 
tively. 



80 



An Experiment 



would appear to describe the performance curve fairly well. Any 
extrapolation of this curve would be unwarranted, however, since the 
maximum performance of 15 problems would be exceeded on a (h3'po- 



lOr- 




= average performonce 
(actual data) 

f\j= cubic regression 

Z';. = -0.65 + 5.71/ -1.98/^+0.212;'^ 

/= linear regression 
Pt = 2.1>7 +0.732/ 



J I 



2 3 4 5 

Test Number 

Fig. 4.1. Performance vs test number. 



thetical) seventh test. However, comparing the cubic with the linear 
trend in Figure 4.1, it would appear that some guesses might be ventured 
about the difficulty of the tests. 



An Experiment 81 

In particular, it might be guessed that test 4 was probably the most 
difficult testji Tests 2 or 6 the easiest. So much for inferences from simple 
appearances of the curve plotted through the observations relative to the 
straight line shown. Furthermore, if an exponential learning curve had 
been considered. Test 5 might well be above the reference curve (the 
exponential) so that the apparent case might be deceptive. 

In order to determine the possible effects of the difficulty of the test 
on the relative performance of the groups, a variance analysis was 
performed on the individual tests, ^ but the results can only be suggestive. 
The highly significant differences in performance appear to be a result 
of the performance on Tests 3 and 5 (in which aspiration level groupings 
are significant) and Test 6 (on which the probability of a difference is 
at least finite) . Finite probabilities of a difference between budget groups 
exist in Tests 2 and 6, and (ah) interactions appear possible on Tests 
3 and 6. 

On Tests 1 and 4 the grouping of the data serves only to increase 
the variance estimate,^ which would suggest that the difference between 
the groups initially was insignificant, and the difficult task, 6, rather than 
accentuating the differences appears to have reduced them. 

4.8. Analysis of Results — Aspiration Level 

In order to investigate in more detail the possible causes of the 
differences in performance noted between groups, the data which were 
acquired on the aspiration levels of the subjects may next be examined. 
The measure of aspiration level which will be used is the subject's 
answer to the question, ''How many problems do you hope to complete 
on Test X?''^ 

The aspiration level on Test 1 was formulated by the subjects with 
incomplete information about the tasks. Group ^ had only the two 



^The "difficulty" of Test 4 was undoubtedly enhanced by other factors: (1) 
subjects who had been unsuccessful on three consecutive trials had no money at all; 
(2) the sequence of budget adjustments produced the higher of the two "adjustments" 
so that the discrepancy between the budget and the perceived possible achievement 
may have been sufficiently large to cause many subjects to "give up;" and (3) a few 
subjects who had formed some estimate of the budget-setting procedure and the 
discrepancy noted in (2) may have been induced by them to use Test 4 to "try it out." 

2As indicated in Appendix 4B. 

^In fact, a result F < 1 is obtained. 

■^It will be recalled that this question was asked only of group /3 and y subjects, 
thus reducing the sample for this discussion to 72. 



82 An Experiment 

sample problems on which to base their aspirations; Group A7 had little 
more/ but the B7, C7, and D7 groups received their budget of 5 prob- 
lems prior to formulating their aspiration levels. ^ Either because the 
sample problems were easier than the test problems or because they were 
perceived as being easier^ — or perhaps because of some other factor — , 
the group jS subjects' average aspiration level was 8.5 problems, com- 
pared with a performance average of 3. Group A7 subjects' average 
aspiration level was 6.6 compared with a performance of 3.2, but the 
subjects who had received the budget of 5 had an average aspiration of 
5.9 compared with a performance of 3.5.^ 

Perhaps even more indicative of the attraction power of the budget 
in the incomplete information situation is the comparison of the number 
of people who chose the number "5" for their aspiration level on the first 
test. Only 1 of 36 Group /3 subjects chose 5, although five chose numbers 
below 5; in group A7, 2 subjects chose 5 (and 3 subjects a number below 
5) ; in the combined B7, C7, and D7 groups, 11 subjects out of 27 chose 5 
but only 1 subject chose a number below 5. It is also interesting to note 
that 6 people in the 13 group chose 15 (maximum possible), and no subject 
in B7, C7, or D7 did so. 

Thus the budget, in the absence of complete information about the 
task, became the aspiration level for more than one-third of the subjects. 



lit is possible that a few of the A7 subjects may have spied the budgets of 5 
received by the other 7 subjects in the same room. In the absence of information, 
the subjects' curiosity in some cases was difficult to control. 

^Thus all of the Group /3 subjects may be considered as a single homogeneous group 
for the first aspiration level determination; B7, C7, and D7 are likewise a homoge- 
neous grouping; but A7 constitutes a third group. 

^Once the solution to a water-jar problem is "seen" the problem usuall}^ seems 
trivial. Since the sample problems were already solved on the instruction sheet, 
the real task — i.e., discovering the solution — had already been performed and the 
subjects may have included only the time they thought necessar}^ to write down 
the solution in their estimates. Although the sample problems may have been easier 
than some of the problems at the end of the test, only a few of the subjects solved 
enough problems on the first test even to attempt the problems which maj^ have been 
more difficult than the sample. 

^The difference between the mean aspiration level of the subjects and the 7 
subjects (excluding A7) was significant at the 1 per cent level. The j3 group was much 
more variable than the combined B7, C7, and D7 groups in its aspiration, and the 
ratio (13 /y) of the sums of squares (adjusted by the appropriate degrees of freedom) 
is significantly greater than 1 at the .1 per cent level. The estimates of the population 
variance from the two samples were a^ = 15.1 and d^ = 1.62 for ,3 and 7 (excluding 
A7) respectively. 



An Experiment 83 

This same information would appear to have eUminated subjects who 
might have otherwise aspired to the maximum (technological optimum) 
score. Finally, it (1) significantly lowered the average aspiration level and 
(2) caused a significant decrease of variability of the aspiration level 
about its average value. ^ A question remains, of course, as to whether a 
budget which was higher than the average of the /3 group aspirations — 
say, a budget of 10 or 12 correctly completed problems — would have 
significantly raised the aspiration level of the y groups. ^ 

Two factors are of interest in comparing the performance of the two 
groups. In spite of the high level of aspiration, the ^ groups performed, 
on the average, .52 problems less than the B7, C7, and D7 combination. 
(It should be observed, however, that the null hypothesis cannot be 
rejected in this case at even the 10 per cent significance level.) The /3 
groups, however, in the absence of an externally imposed factor, ap- 
peared in some degree to reflect their abilities, at least relative to one 
another, in their aspiration level determination, while the groups which 
were influenced by the budget did not. 

This last phenomenon is exhibited by the correlation estimates. The 
coefficient of correlation between aspiration level and performance for 



^Actually, it was possible to reject the null hypothesis (^ = 5) in either group even 
at the 1 per cent level. For the sake of exposition, 5 per cent confidence limits for 
the two groups were : 

7.22 < M < 9-86 for the /3 group 

5.42 < Ai < 6.44 for B7, C7, and D7 groups combined 

Thus it is apparent that the budget presented in the absence of other information 
influenced, but did not necessarily become, the aspiration level. The obvious difficulties 
which accrue from aggregation here require further study beyond the scope of this 
thesis. 

2It is of interest here to compare these results with those of Chapman and Volk- 
mann (7). Their experiment involving aspiration levels in the absence of information 
about the task (summarized on pp. 44-45 of this thesis) showed a reluctance on the 
part of the subjects to aspire even to the average performance of "unselected WPA 
workers." On the other hand, the subjects of the experiment presented here aspired 
to a level on the average higher than the stated goal which corresponds to the Chap- 
man and Volkmann group norm insofar as it is the only figure which the subjects 
can use as a reference point. Although this difference between the subjects of the 
two experiments can be explained entirely in terms of differences in subjects and 
experimental procedures, the possibility exists that in spite of attempts by the 
experimenter, subjects will develop some intuitive (if irrational) performance esti- 
mates prior to their first encounter with the task. These a priori estimates appear 
to be fairly easily changed by experience with the task but resist change from an 
external source, at least to some extent. 



84 An Experiment 

the /5 subjects was .436 (significant at the 1 per cent level), ^ while for 
Bt, Cy, and D7 subjects, the correlation was negative ( — .063) but not 
significantly different from 0. Group A7 had an average performance 
of 3.2. This was closer to /3 than to the other 7 groups and also resembled 
the (3 group in its aspiration performance correlation of .318 (significant 
at the 5 per cent level) and its variance {a^ = 15.3). 

It may be then hypothesized that, in the absence of complete infor- 
mation about the task, some ranking of aspiration levels relative to 
maximum and minimum values occurs and that this ranking reflects 
the abilities of the individuals — at least relative to one another — and, 
finally, that the imposition of an external goal interferes with this 
ranking. The alternate hypothesis, viz., that the aspiration level which 
is set without reference to an external goal has a greater effect on 
performance than one which is seen, at least partially, to be forced upon 
the individual,^ does not receive much support since 7 performance was 
superior to (3 performance. On the other hand, the inferiority of /3 
performance may support the existence of a point of maximum stress 
hypothesized above (Postulate (iii.a) of Chapter 2), but it must be 
admitted that the evidence is not conclusive. Furthermore, in viewing 
these results the problem of bias due to the disparity of educational level 
of the subjects noted above must be considered. 

It should be observed that possible interference with aspiration level 
determination, due to learning about the next test before it is admin- 
istered, may occur, in that the subjects appeared to become more adept 
at solving the problems as the test trials progressed (at least for the 
number of tests used). On the other hand, it is unlikely that the in- 
cremental gain in information about the task, which would aid in deter- 
mination of the aspiration level, is significant on any test other than the 
first. ^ Hence it is possible to group the aspiration levels from Tests 2 to 6 
together as ''informed" levels. 

There appeared to be relatively little difficulty in dislodging the 



iThis result is consistent with the results of Chapman and Volkmann (7) (dis- 
cussed in Chapter 3 of this thesis) . They, too, found that in their control group (which 
had no external information other than minimum and maximum possible scores'! the 
subjects' performance and aspirations were positively and significantly correlated. 

^This seems to be the import — as nearly as can be ascertained — of the sugges- 
tions and comments made by Argj^ris (2). 

31.6., it is unlikely that the subject possesses any more information at the end of 
Test 3 about Test 4 (relative to his ability) than he had at the end of Test 2 about 
Test 3 (although he probably has more information about Test 4, relative to his 
ability than he had at the end of Test 2). 



An Experiment 85 

subjects from the initial aspiration level once the first test had been 
performed. Of the 72 subjects whose aspirations were requested, only 9 
duplicated their original choices on the second test. Of the 9, two were y 
subjects whose choice was reinforced by having their budgets coincide 
with the previous aspiration level. Three others maintained the same 
aspiration level throughout.^ 

The small number of duplications and the attached mitigating 
statements tend to indicate that (since the expected value of the number 
of repetitions, assuming the choices independent, is yg- X 72 = 4.8) the 
first aspiration level choice does not substantially affect subsequent 
choices, except in cases of extreme "rigidity" for which the circumstances 
of the first choice are vital. Since the cases in which no change at all 
occurred in the aspiration level numbered only 3, or 4 per cent of the 
sample, it is not likely that the results of the study of the 'informed" 
aspiration level would have been drastically changed if the aspiration 
level determination for the first test would have been eliminated. 

Uncovering or discovering causal factors in a statistical analysis 
draw^n from such a model design is a somewhat tricky business. For this 
purpose, the following suggested procedure might be employed in this 
case. To distinguish causal relationships that may exist with regard to 
the formulation of the aspiration level and its subsequent effect on 
performance, two types of discrepancies might be distinguished for 
analysis. Lewin et al. (37b) have defined the difference between the 
aspiration level for a test and the score on the test as the ''achievement 
discrepancy," while the difference between the aspiration level and the 
performance on the previous test is called the ''goal discrepancy." In a 



^This phenomenon has been investigated thoroughly by Rotter (67) and others 
using the "Level of Aspiration Board" which involves a task of rolling a steel ball 
along a groove. Terming this absence of shifts the "rigid pattern," Rotter states: 
"Rigidity here is to be thought of as primarily an avoidance response — a way of 
avoiding decisions or situations in which commitment may lead to a mistake or 
punishment." ((67), p. 322.) Six other subjects shifted after the first test but 
retained the level chosen for the second test thereafter; these would also fit Rotter's 
definition of the "rigid pattern." It is interesting to note that all cases of rigidity 
occurred in the fi groups and Group A7. Assuming an average occurrence of rigidity 
to be 7% = I for the entire group, the probability that all 9 would occur in 5 groups 
(45 subjects), or that 3 groups (27 subjects) would contain no cases of rigidity is 
^2^/^2 7 or about .010. This assumes a finite population. If, alternately, an infinite 
population were assumed with rigidity occurring in I of the population, then the 
estimate of the chance probability of drawing 27 without rigidity would be (l)^^ ^ 
.027. Hence it must be concluded from these data and for this experiment that an 
externally imposed goal prior to the formation of aspirations significantly reduces 
"rigidity." 



86 An Experiment 

situation in which the amount of performance improvement per trial is 
small, the difference between the two discrepancies will be small and is 
usually ignored. 1 If a continuous model is assumed (e.g., the model of 
Chapter 2) the discrepancies must be identical. 

The model of causality assumed for present purposes is shown in 
Figure 4.2. It will be noted that, as in the mathematical model, the 



f\ f\ 

!y^ ^ 'y^ 

Pf = performance on test / 

Of = aspiration level for test f 

bf = budget level for test / 

>■ = management action (predetermined) 

^= direction of causality in subjects 

(department head's) behavior (under 
investigation ) 

Fig. 4.2. Causality relations. 

budget is assumed to work through the aspiration level to performance 
rather than directly. 

Studies of the goal discrepancy in the absence of a budget have shown 
that its magnitude can be reliably predicted from an independent study 
of personality. 2 In the absence of a priori indications of differences in 
personality between groups^ it may be concluded that if the between 
group differences in goal discrepancy are significant, it will be possible 
to establish a causal relationship leading from the type of budget and/or 
aspiration level determination to the goal discrepancy. Since the 
previous performance is an exogenous variable at the time of aspiration 
level determination, the causal relationship can be extended to the 
aspiration level. 



iSee, for example, Rotter (67), Chapter II, particularly pp. 128-130, and Chapter 
VIII, pp. 313-326. 

nud. 

^Actually, the extremes of the subject classifications were largel}'' confined to the 
a groups, and noting Table 4A, the jS and y subgroups differed for the most part in 
terms of whether the 3 or 4 seniors in the group were studying management in the 
regular curriculum or on an elective basis. 



An Experiment 87 

The goal discrepancy analysis of variance ^ shows that the differences 
between budget groups are highly significant as are the three-way 
interactions. The latter can be accounted for by the budget setting 
procedure. The goal discrepancy followed a strong regular pattern in the 
B7, C7, and D7 groups which was absent in the others.^ The fact that 
the test effect was not significant would tend to vahdate the assumption 
made above, viz. that the amount of additional information received 
about the task on tests other than the first is small. 

Returning to Table 4.3, it is seen that the average goal discrepancy 
exceeded the "ideal" goal discrepancy (which would occur if management 
could control the aspirations exactly as a< = ht) in the A groups and B7 
but was below the "ideal" elsewhere. The amount of the difference 
between "ideal" and actual discrepancy increased with the severity of 
the budget in the B7, C(3, and D7 groups, but decreased in the B7, C/3, 
and D|S groups. Taken together, these statements would imply that the 
more explicit information at the time of formulation of the aspiration 
level tended to mitigate rather than accentuate the differences between 
budgets. Unfortunately, however, significant (ab) interactions, which 
would be required for validity of this conclusion, are not present. 

The achievement discrepancy analysis of variance^ showed highly 
significant test and budget effects. Since the test effect of the goal 
discrepancy is not significant, the ranking of the achievement dis- 
crepancy can be assumed to be a reasonably accurate measure of the 
differences in difficulty between tests. The ranking indicates the order 
of difficulty as 4, 3, 2, 5, 6, which is in accord with the comments in 
Section 4.7. This achievement discrepancy is comparable with the dis- 
crepancy which is considered to be a measure of stress in Chapter 2 (i.e., 
frustration is evoked in the attempt to perform at a level of aspiration 
(already set) above ability, not in the process of setting the aspiration "*). 

Comparing Tables 4.3 and 4.1, it may be noted that the average test 
scores from 2 — 6 preserve the same ranking by subgroups as the scores 



iTable 4B.8 of the Appendix. 

2As will be recalled, the budgets were set by adding a number (1 or for Group B, 
2 or 1 for Group C, etc.) to the previous score. The higher of the two numbers always 
occurred on tests 2, 4, and 6. Subjects in Groups B7, C7, and D7 (having seen the 
budget) often set their aspirations at, or a fixed number below or above, the budget. 
Thus the goal discrepancies of these subjects followed the high-low pattern of the 
alternating budgets, while this effect did not (and would not be expected to) prevail 
in other groups. 

^Table 4B.9 of the Appendix. 

4Ruch (68), Chapter 7. 



88 



An Experiment 





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^ B- 












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tn if, 




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r- 




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ca 






to 


>> CD 
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— 




m 


^ « 


CC 


^ X 


CE 


S^' oc 


Cj 






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(D 


o rrl .2i o 


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epanc 
Level 
nt Dii 
ce, Te 


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_c; 






+ II 1 II 




+ II 1 II 




+ II 1 II 


^ 



An Experiment 89 

for 1 — 6 except that the insignificant difference between C7 and C/3 is 
ehminated. It should be noted that stress in D exceeded stress in C and 
A, although performance was ranked A > C > D > B. The ranking 
of budget groups and D7, viz. D7 > A > C > B, also does not quite 
correspond with the stress (achievement discrepancy) ranking for those 
groups, in that stress in D7 is less than stress in A. However, the rela- 
tionship between stress and performance may have some validity since 
the functional dependence of performance and achievement discrepancy 
produces an influence counter to the ordering shown, and a somewhat 
altered measure of stress might produce more reliable results. The D(3 
group, and/or some of the subjects in it, may exhibit the phenomenon 
of stress exceeding some maximum limit. (See postulates (iii) of the 
model in Chapter 2). Although not conclusive, the results suggest that 
the effect of higher stress, up to a point, is to increase performance. 

The study of the achievement discrepancy is somewhat disappointing 
insofar as no definite conclusions can be drawn other than an estimate 
of the difficulty of the tests. On the other hand, if the achievement 
discrepancy is a measure of stress, then a task which is considerably more 
difficult than usual may cause stress to exceed some tolerable value. ^ 
Furthermore, this difficult task may appear (as it did here) without 
either the intention or the knowledge of management. 

4.9. Conclusions 

The results of the experiment have shown that performance in a 
situation where the attainment of a goal is rewarded and its nonattain- 
ment penalized is significantly affected by the type of budget chosen, 
the conditions of administration, and the way in which aspiration levels 
for the task are determined. 

The experimental results indicate that an "implicit" budget (where 
the subject is not told what goal he must attain) produces the best 
performance, closely followed by a ''medium" budget and a ''high" 
budget. The "low" budget, which was the only one which satisfied 
the criterion of "attainable but not too loose," resulted in performance 
significantly lower than the other budget groups. 

However, there is a strong interaction effect between budgets and 
the aspiration level determination grouping. The group of "high" budget 
subjects who received their budgets prior to setting their aspiration levels 
perfomed better than any other group, whereas the "high" budget group 



^See remarks in Sections 2.3 and 2.4 regarding stress. 



90 An Experiment 

who set their aspirations before receiving the budget were the lowest 
performers of any group. 

An hypothesis which might satisfactorily explain this phenomenon is 
as follows: The high performing group formed its aspirations with the 
high budget levels in mind, while the low performing group rejected 
the high budget after forming aspirations with relation to their last 
performance. However, aspiration level data indicate that the low 
performing group had a much higher goal discrepancy so that, if any- 
thing, their goals were closer, on the average, to the budget than were 
the high performers. 

The low performing group also had a very high achievement dis- 
crepancy score. If achievement discrepancy is interpreted as a measure 
of stress, this would give rise to an alternate hypothesis — viz. that the 
stress, at least for some subjects, was so high that they may have been 
"discouraged" and may have ceased to try to improve perform- 
ance. 

A major difficulty is involved in the use of achievement discrepancy 
scores. This results from the functional dependency of such scores upon 
performance. For example, a constant aspiration level would produce a 
negative correlation between achievement discrepancy and performance 
(ignoring the trivial case where performance equals aspiration level 
throughout). This conceptual deficiency of the current state of psycho- 
logical theorizing makes it difficult to establish a causal relationship 
between this discrepancy and performance — as hypothesized in this 
model. It is, however, significant that the ordering of the stress (achieve- 
ment discrepancy) corresponded fairly closely to the ordering of perform- 
ance in spite of the opposing effect of the functional dependency. 

It is also observed from the analysis that the size of the achievement 
discrepancy can be affected significantly by the size of the budget, 
a result which is consistent with the requirements for the validity of 
postulate (ii) in the model of Chapter 2. 

The investigation of the goal discrepancy also indicated a significant 
budget effect, and this again tends to corroborate postulate (ii), since 
in the continuous model the two discrepancies are indistinguishable. 
A somewhat surprising (even though not statistically significant) effect 
is also present. This is the tendency of the groups who formed their 
aspiration levels without knowledge of the budget to come closer to 
their budget than is true for the group which had the advantage of the 
budget information supplied to them. A possible explanation is that 
a moderate departure from the budget can come about gradually in 
the groups that form aspirations first, while a departure in the other 



An Experiment 91 

groups (since the budget is associated with reward) is Ukely to be a large 
shift downward. 

The types of aspiration levels, under the procedures used, did play 
a part in performance differences. This was to be expected. But the fact 
that it did not play a part in the discrepancies which occurred requires 
some explanation. One possibility is that the greater variability of the 
aspirations noted in the aspirations of the ,3 subjects (who did not have 
the budget when forming their aspirations), in spite of roughly the same 
average aspirations, may have produced a greater number of discouraged 
subjects (who had aspired to extremely high levels). In the group whose 
aspiration levels tended to remain close to the budget, there were 
relatively few extremely high or low discrepancies. Hence this group 
would tend to exhibit high but not intolerable levels which, according 
to postulates stated earlier, would lead to high performance as well. 

Although not conclusive on the point, the study does shed some light 
on participative schemes of budgetary management insofar as these 
are connected to aspiration levels. The group which determines its 
aspiration level first, in the experimental situation, is closest to the 
solution proposed by MacGregor (54). He suggests that the department 
head should plan his budget and then take it to his supervisor who will 
give him his budget based on his estimate. The experimental data raiee 
some questions as to the universal validity of this recommendation, for 
under the experimental situation if ''management" decides on a "high" 
(performance) budget, its use of MacGregor's participation plan coin- 
cides with the worst possible result. On the other hand, it would 
probably help performance in a ''low" budget situation. 

This summary of the findings may now be concluded by a few 
observations on the causal connection between stress and performance. 
As already noted, the experiment did help to separate and distinguish 
between goal discrepancy and achievement discrepancy. The best that 
the data and subsequent analyses will bear on the subject of "stress" 
suggests only the possible use of achievement discrepancy as either a 
surrogate or a direct measure of stress. The value of the achievement 
discrepancy as a measure of stress is dubious and the effect of stress 
and/or achievement discrepancy on performance requires further 
documentation before such usage is fully warranted. 



APPENDIX 4A 

DETAILS OF THE EXPERIMENTAL DESIGN 

4A.1. Task Selection 

The capacities for the three jars were chosen from a table of random 
numbers, 1 to 50 inclusive, eliminating duplications (in the same prob- 
lem) when they occurred. A '^solution" was then selected at random 
from a prepared set of nine possible solutions. Some selection of a parent 
population was necessarily and unavoidably involved at this stage. The 
criterion employed was an initial judgment that each set to be sampled 
was fairly close in order of difficulty. As a precaution, preliminary or pilot 
tests were conducted, but these could not be undertaken on a sufficient 
scale to empirically compare the results of the selection process with the 
criterion of equal difficulty of the tests. In any event the population 
(of solutions) so chosen was used for the random draws and a requirement 
determined accordingly. 

In more detail this phase of the design was as follows: the solution 
for problem 1 was chosen at random from groups 1, 2, and 3 of Table 4A.1. 
In like manner the solution for problem 2 was chosen from groups 2, 3, 
and 4, etc. In this table the letters A, B, and C refer to the largest, 
middle, and smallest jars, respectively. (In the actual test A, B, and C 
are listed as they occurred in the table of random numbers.) If the 
solution chosen produced an impossible requirement for a set of capac- 
ities (e.g., the requirement 2 A cannot be obtained if A > B + C), 
another solution was chosen at random from the remaining eight, and 
so on until the desired test battery was complete. ^ 



^In every case at least one of the solutions possessed a feasible requirement* 
although there was no guarantee of this. 

93 



94 Details of the Experimental Design 

TABLE 4a.1. required AMOUNTS IN TERMS OF JAR CAPACITIES (a > B > C) 

LISTED IN APPROXIMATE ORDER OF DIFFICULTY.* 

Group 1 A + B + C, 2B, 2C 3 

Group 2 A - B, A - C, B - C 3 

Group 3 2B + C, A + 2C, B + 2C 4 

Group U A-B+C, A + B-C, 2A 4 

Group 6 A - 2C, 3C, 3B(A < 2B) 5 

Group 6 A - 2B, 2B - A, B - 2C 5 

Group 7 2C - B, 2B - C, 2C - A 5 

Group 8 B + C - A, 2A - B(A < 2B), A - B - C 5 

Group 9 2A - C, 2A - B(A > 2B), 3B(A < 2B) 6 

Group 10 A + 3C, B + 3C, 3B + C 6 

Group 11 2B + 2C, A + B - 2C, A - B + 2C 6 

Group 12 B + 2C - A(A < B + C), 2B + C - A(A < 2B), 

A - 2B + C(A < 2B) 6 

Group 13 4C(A > 2C) 4B(A < 4B) A - 3C 7 

Group U A - 3B, B - 3C, 3C - B 7 

Group 15 3B - A, 2B - 2C, 2A - 2C 7 

Group 16 2A - 2B(A < 3B + C), A - 2B - C, A - B - 2C 7 

Group 17 B + 2C - A(A > B + C), 2B + C - A(A < 2B), 

A - 2B + C(A < 2B) 7 

Group 18\ 2A - B - C, 2A - 2B(A < 3B), 3B - C(2B - C < A) 7 

*Items in parentheses denote restrictions on the jar capacities which allow the 
problem to be solved in the indicated number of steps — restrictions for the existence 
of a solution are omitted. 

fShown for completeness; all other problems require 8 or more steps. 

The selection process was intended to give a rough ordering (in terms 
of increasing difficulty) from the first to the last problem on any test. 
Actually, it is almost impossible to determine a priori a precise ordering, 
even if it were deemed desirable, without conducting a pre-trial at least 
as elaborate as the experiment itself. For example, a problem whose 
solution is simple 4C probably presents less difficulty than one involving 
steps leading to a more complicated expression such as B + 2C — A or 
even B + C — A. Note that this may be true even though the steps 
required may be more numerous to reach 4C than one of the more 
complex expressions.^ To state the matter differently, the number of 
steps involved to reach a solution is not precisely related to logical (or 
rather, psychological) difficulty. It is interesting in this connection to 



^Actually, subjects had a great deal of difficulty with a problem whose solution 
was 3C, C = 27, probably due to the effect of EinsteUung (set) investigated by 
Luchins (52) using these problems. Since EinsteUung depends on experience with 
previous problems, ordering appears as a factor in determining difficulty. 



Details of the Experimental Design 95 

observe, for example, that the statistical analysis seems to show that 
the subjects tended to experience greater ''difficulty" in reaching a 
solution as the capacity of the jars was increased — a psychological 
rather than a logical distinction being involved here. 

Another illustration of possible sources of disturbance to the intended 
ordering is provided by reference to a lack of uniqueness for the solutions 
to some problems. For example, for the set of jars {A = 37, B = 18, 
C = 3) the solution B — 2C, or a requirement of 6, was chosen at random. 
This solution requires 5 steps, but the simpler solution 2C requires only 3. 
Since it took some time to discover the existence of these simpler prob- 
lems toward the end of the test, the discovery was considered part of the 
learning process and no attempt was made to eliminate or reorder the 
problems on the basis of alternate solutions.^ 

4A.2. Sample Documents 

This section of the appendix is included to give the reader a better 
idea of precisely what the subjects encountered in the experiment. 
It is also included in the hope that data on similar tests, which would 
be comparable to these will be of aid to other experimenters. 

The foms are filled out as they would be given to a Group B subject 
who attained his budget of 5 on Test 1 and a Group C7 student who 
attained a score of 2 on Test 1, and an Aa subject and an A7 subject 
who did not attain the Test 1 budget. 



^Experience with the tests indicated that this would have been a virtually impos- 
sible task. Subjects discovered as many as six solutions to some of the problems 
including, in several cases, simpler solutions than the experimenter had discovered 
indicating that he, too, exhibited Einstellung. 



PROBLEM-SOLVING EXPERIMENT 

Instructions 

In each problem you have three water jars, A, B, and C, with different 
capacities. With these jars you are expected to measure out the required amount 
of water without approximating. You may fill any of the jars from a spigot, empty 
any of them into the sink, or fill one jar from another jar. After a jar is filled 
from the spigot it must be filled to capacity, and after a jar is emptied into the 
sink it must have no water in it. After one jar is filled from another, either the jar 
which has been filled must be filled to capacity, or the jar from which water is 
poured must have no water. 

For each problem you will be given the jar capacities and the required 
amount. The required amount must be the total amount of water in the three jars 
at the end of the problem. You are to answer the problem b}' filhng in the 
appropriate blanks. Use only one line for each step. 







Contents 






from .... 


A 


B 


c 


Fill .... 


3 








Fill C 


from A 


1 





2 


Fill .... 


from .... 


1 









Sample Problem 1: Capacities in quarts; A = 3, B = 7, 
C = 2. Required 1 quart. 
Step 

1. Fill A Empty .... 

2. Fill .... Empty .... 

3. Fill .... Empty C 

Clearly, this is not the only solution to this problem. One alternate would be : 
Fill B, fill A from B, empty A, fill A from B, empty A. Either solution would be 
correct. In more complex problems it may be desirable to keep count of the 
number of quarts in each jar in the space at the right, as shown. This is not 
necessary, and only the filling-in of the blanks in the steps will be checked. (The list 
of contents will not be checked.) 

A more complicated sample problem is shown below. 



Sample Problem 2 


.• Capacities 


A = 27, 


B = 31, C 


Required 18. 








Fill B 


Empty .... 


Fill .... 


from .... 


Fill .... 


Empty .... 


Fill C 


from B 


Fill .... 


Empty C 


Fill .... 


from .... 


Fill .... 


Empt}^ .... 


Fill A 


from B 


Fill B 


Empty .... 


Fill .... 


from .... 


Fill .... 


Empty .... 


Fill A 


from B 


Fill .... 


Empty A 


Fill .... 


from .... 



17. 



Contents 



A 


B 


c 




31 






14 


17 




14 




14 






14 


31 




27 


IS 
IS 





Exhibit 4A.1. Instructions (all groups) 
96 



You will be given six tests of equal difficulty, each containing 15 water jar 
problems. You will have seven minutes for each test. You are to complete 
as many problems as you can in the seven minutes allowed. A goal will be set 
on each test by which your performance will be judged. If you attain the goal 
(that is, complete the required number of problems correctly) you will be 
rewarded; if you complete fewer problems than the goal, you will be penalized. 
The reward on the six tests will be that you will receive $1.00. If you do not 
attain your goal on any of the six tests, you must pay back $1.00 as your penalty. 

You will be given $3.00 at the start of the experiment. Should you have no 
money left at the start of any test, you can still earn the reward, but you will not 
pay the penalty. If you attain the goal on all six tests you will go home with $9.00. 
If you do not attain the goal on any test you lose nothing. 

Before beginning Test 1 [you will be asked some questions on cards that will 
be passed out to you and] you will receive additional information.* During a 
three-minute break between tests your performance will be evaluated with respect 
to the goals which have been set for you and the rewards and penalties will be 
administered. [You will then be given additional information and asked some 
more questions. The answers to these questions will not be used in the deter- 
mination of the goals. When you have answered the questions, turn the card over 
on the table. Once you have turned the card over, you will not be allowed to 
change your answers.] 

At the end of the experiment you will be asked to sign a receipt for the money 
you have earned. // you do not complete the experiment [or do not answer all of 
the questions on the cards which have been passed out to you], you will not be allowed 
to take your earnings from the room. 

YOU MUST NOT REVEAL THE NATURE OF THE EXPERI- 
MENT TO OTHERS WHO MAY PARTICIPATE AS SUBJECTS 
THERE MUST BE NO COMMUNICATION BETWEEN SUB- 
JECTS DURING THE EXPERIMENT 



*The form shown is for /3 and y subjects. The Group a had instructions with 
the portions in brackets deleted. 



Exhibit 4A.1 (cont.). Instructions (all groups) 



97 



Problem 1: A 
Step 
1. 
2. 
3. 
4. 
5. 



9. 
10. 



WATER JAR EXPERIMENT 

21, B = 43, C = 28. Required 15. 



Fill .. 


Empty ,. 


Fill .. 


Empty .. 


Fill .. 


Empty .. 


Fill .. 


Empty .. 


Fill .. 


Empty .. 


Fill .. 


Empty .. 


Fill .. 


Empty .. 


Fill .. 


Empty .. 


Fill .. 


Empty .. 


Fill .. 


Empty ... 



Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 



Test 2 

Jar Contents 

ABC 



from 
from 
from 
from 
from 
from 
from 
from 
from 
from 



Problem ^; A = 31, B = 8, C = 41. Required 57. 



Step 
1. Fill 



2. 


Fill 




3. 


Fill 




4. 


Fill 




5. 


Fill 




6. 


Fill 




7. 


Fill 




8. 


Fill 




9. 
0. 


Fill 
Fill 





Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 



Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 



from 
from 
from 
from 
from 
from 
from 
from 
from 
from 



Problem 3: A 
Step 



47, B = 30, C = 14. Required 42. 



Fill 
Fill 
Fill 
Fill 
Fill 

6. Fill 

7. Fill 

8. Fill 
Fill 
Fill 



9. 
10. 



Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 



Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 



from 
from 
from 
from 
from 
from 
from 
from 
from 
from 



Exhibit 4A.2. Sample test (all groups) 
98 



Problem 4: A = 30, B = 27, C = 17. Required 40. 



Step 
1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 



Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
Fill 
10. Fill 



Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 



Test 2 

Jar Contents 

ABC 



... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 



Problem 5: A = 30, B = 27, C = 36. Required 33. 



Step 
1. 



Fill 



2. 


Fill .... 


3. 


Fill .... 


4. 


Fill .... 


5. 


Fill .... 


6. 


Fill .... 


7. 


Fill .... 


8. 


Fill .... 


9. 


Fill .... 


0. 


Fill .... 



Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 
Empty 



... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 


.... Fill .. 


from .... 


... Fill .. 


from .... 


... Fill .. 


from .... 



Problem 6: A 
Step 



11, B = 25, C = 24. Required 26. 



Fill 
Fill 
Fill 
Fill 
Fill 

6. Fill 

7. Fill 

8. Fill 

9. Fill 
10. Fill 



Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 


Empty .. 


Fill .. 


from .... 



Exhibit 4A.2 (cont.). Sample test (all groups) 
99 













Test 2 


Problem 7: A 


= 50, B = 49 


C = 19. 


Required 18. 


Jar Contents 


Step 










ABC 


1. 


Fill .. 


Empty .. 


Fill . 


from .... 





2. 


Fill .. 


Empty .. 


Fill . 


from .... 





3. 


Fill .. 


Empty .. 


Fill . 


from .... 





4. 


Fill .. 


Empty .. 


Fill . 


from .... 





5. 


Fill .. 


Empty .. 


Fill . 


from .... 





6. 


Fill .. 


Empty .. 


Fill . 


from .... 





7. 


Fill .. 


Empty .. 


Fill . 


from .... 





8. 


Fill .. 


Empty .. 


Fill . 


from .... 





9. 


Fill .. 


Empty .. 


Fill . 


from .... 





10. 


Fill .. 


Empty ... 


Fill . 


from .... 





Problem 8: A 


= 5, B = 45, 


C = 40. 


Required 85. 




Step 












1. 


Fill .. 


Empty ... 


Fill . 


from .... 





2. 


Fill .. 


Empty ... 


Fill . 


from .... 




3. 


Fill .. 


Empty ... 


Fill . 


from .... 




4. 


Fill .. 


Empty ... 


Fill . 


from .... 





5. 


Fill .. 


Empty ... 


Fill . 


from .... 




6. 


Fill .. 


Empty ... 


Fill . 


from .... 


.... 


7. 


Fill .. 


Empty ... 


Fill . 


from .... 


.... 


8. 


Fill .. 


Empty ... 


Fill . 


from .... 


.... 


9. 


Fill .. 


Empty ... 


Fill . 


from .... 


.... .... 


10. 


Fill .. 


Empty ... 


Fill . 


from .... 





Problem 9: A -- 


= 43, B = 36, 


C = 23. 


Required 33. 




Step 












1. 


Fill .. 


Empty ... 


Fill. 


from .... 


.... .... 


2. 


Fill .. 


Empty ... 


Fill. 


from .... 


.... .... 


3. 


Fill .. 


Empty ... 


Fill . 


from .... 




4. 


Fill .. 


Empty ... 


Fill. 


from .... 




5. 


Fill .. 


Empty ... 


Fill. 


from .... 


.... .... 


6. 


Fill .. 


Empty ... 


Fill. 


from .... 


.... 


7. 


Fill .. 


Empty ... 


Fill .. 


from .... 


.... 


8. 


Fill .. 


Empty ... 


Fill .. 


from .... 


.... 


9. 


Fill ,. 


Empty ... 


Fill .. 


from .... 




10. 


Fill .. 


Empty ... 


Fill .. 


from .... 





Exhibit 4A.2 (cont.)« Sample test (all groups) 
100 



^ 













Te8i2 


Problem 10: A = 


= 35, B = 18 


, C = 33 


. Required 34. 


Jar Contents 


Step 










ABC 


1. 


Fill .... 


Empty .... 


Fill .. 


from .... 




2. 


Fill .... 


Empty .... 


Fill .. 


from .... 




3. 


Fill .... 


Empty .... 


Fill .. 


from .... 




4. 


Fill .... 


Empty .... 


Fill .. 


from .... 




5. 


Fill .... 


Empty .... 


Fill .. 


from .... 


.... .... 


6. 


Fill .... 


Empty .... 


Fill .. 


from .... 


.... .... 


7. 


Fill .... 


Empty .... 


Fill .. 


from .... 


.... 


8. 


Fill .... 


Empty .... 


Fill .. 


from .... 


.... 


9. 


Fill .... 


Empty .... 


Fill .. 


from .... 


.... 


10. 


Fill .... 


Empty .... 


Fill .. 


from .... 




Problem 11: A = 


= 12, B = 1, 


C = 45. 


Required 48. 




Step 












1. 


Fill .... 


Empty .... 


Fill .. 


from .... 




2. 


Fill .... 


Empty .... 


Fill .. 


from .... 


.... .... 


3. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... .... 


4. 


Fill .... 


Empty .... 


Fill .. 


from .... 


.... .... 


5. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... .... 


6. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... 


7. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... 


8. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... 


9. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... 


10. 


Fill .... 


Empty ... 


Fill .. 


from .... 




Problem 12: A = 


= 2, B = 42, 


C = 40. 


Required 36. 




Step 












1. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... 


2. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... 


3. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... 


4. 


Fill .... 


Empty ... 


Fill .. 


from .... 


.... .... .... 


5. 


Fill .... 


Empty ... 


Fill. 


from .... 


.... .... .... 


6. 


Fill .... 


Empty ... 


Fill. 


from .... 


.... 


7. 


Fill .... 


Empty ... 


Fill. 


from .... 


.... 


8. 


Fill .... 


Empty ... 


Fill. 


from .... 


.... 


9. 


Fill .... 


Empty ... 


Fill. 


from .... 


.... 


10. 


Fill .... 


Empty ... 


Fill . 


from .... 






Exhibit 4A.2 (cont.). Sample test (all groups) 
101 







Test 2 


Problem 13: A 


= 23, B = 48, C = 19. Required 76. Jar Contents 


Step 




ABC 


1. Fill .. 


Empty .. 


.. Fill ... from .. 




2. Fill .. 


Empty .. 


.. Fill .... from .. 


. 


3. Fill .. 


Empty .. 


Fill .... from .. 




4. Fill .. 


Empty .. 


.. Fill .... from .. 


. 


5. Fill .. 


Empty .. 


.. Fill .... from .. 


. 


6. Fill .. 


Empty .. 


.. Fill .... from .. 


. 


7. Fill .. 


Empty .. 


.. Fill .... from .. 


.... .... 


8. Fill .. 


Empty .. 


.. Fill .... from .. 





9. Fill .. 


Empty .. 


.. Fill .... from .. 


.... .... 


10. Fill .. 


Empty .. 


.. Fill .... from .. 





Problem U: A 


= 42, B = 22, C = 31. Required 40. 


Step 






1. Fill .. 


Empty .. 


.. Fill .... from .. 


.... .... 


2. Fill .. 


Empty .. 


Fill .... from .. 


.... .... 


3. Fill .. 


Empty .. 


.. Fill .... from .. 




4. Fill .. 


Empty .. 


.. Fill .... from .. 


.... .... 


5. Fill .. 


Empty .. 


Fill .... from .. 


.... .... 


6. Fill.. 


Empty .. 


.. Fill .... from .. 


.... .... 


7. Fill .. 


Empty .. 


Fill .... from .. 


.... .... 


8. Fill .. 


Empty .. 


Fill .... from .. 


.... .... 


9. Fill .. 


Empty .. 


.. Fill .... from .. 


.... .... 


10. Fill .. 


Empty .. 


.. Fill .... from .. 





Problem 15: A 


= 27, B = 9, C = 8. Required 1. 




Step 






1. Fill . 


Empty .. 


Fill .... from .. 


. 


2. Fill .. 


Empty .. 


Fill .... from .. 


.... .... 


3. Fill .. 


Empty .. 


.. Fill .... from .. 




4. Fill .. 


Empty .. 


.. Fill .... from .. 




5. Fill .. 


Empty .. 


Fill .... from .. 


.... .... 


6. Fill .. 


Empty .. 


.. Fill .... from .. 


.... 


7. Fill .. 


Empty .. 


Fill .... from .. 


.... 


8. Fill .. 


Empty .. 


.. Fill .... from .. 


.... 


9. Fill .. 


Empty .. 


.. Fill .... from .. 




10. Fill .. 


Empty .. 


Fill .... from .. 





Exhibit 4A.2 (cont.). Sample test (all groups) 
102 



Based on your experience on Test 1 and the knowledge that Test 1 and 
Test 2 are of equal difficulty, please answer the following questions. 

How many problems do you personally hope to complete on 
Test 2? 

Do you believe that you should be penalized if you do not com- 
plete as many problems as you hope to? 

(yes or no) 
If your answer to the last question was "no," at what level do you think 
we should set the goal for your performance? In other words, how many 
problems do you think we should require you to complete in order to receive 
the reward? 



Exhibit 4A.3. Sample aspiration level determination questionnaire 

(Group ^) 



The goal has been set by which 
your performance on Test 2 will 
be judged. If you solve the goal 
number of problems or more 
correctly, you will receive $1.00. 
If you solve fewer problems than 
the goal you must pay back $1 .00. 
You will not be told what this 
goal is. Work as hard as you can 
on this test. 



The goal by which your perform- 
ance on Test 2 will be judged is 6 
problems. If you solve this num- 
ber of problems or more correctly, 
you will receive $1.00. If you 
complete fewer problems than the 
goal, you must pay back $1.00. 



Exhibit 4A.4. Sample budget 
(Groups Aa, Ba) 



Exhibit 4A.5. Sample budget 
(Groups Ba, Ca, Da, B^, C^, D^) 



The goal by which your performance on Test 2 will be judged is 4 problems. 
If you solve that number of problems or more correctly, you will receive 
$1.00. If you complete fewer problems than the goal, you must pay back 
$1.00. 

Based on your experience on Test 1 and the knowledge that Test 1 and 
Test 2 are of equal difficulty, please answer the following questions. 
How many problems do you personally hope to complete on 

Test 2? 

Do you believe that you should be penalized if you do not com- 
plete as many problems as you hope to? 

(yes or no) 
If your answer to the last question was "no," at what level do you think 
we should set the goal for your performance? In other words, how many 
problems do you think we should require you to complete in order to receive 
the rew^ard? 



Exhibit 4A.6. Sample budget — aspiration level combination 
(Groups By, Cy, Dy) 

103 



A goal has been set by which your performance on Test 2 will be judged. 
If you solve the goal number of problems or more correctly, you will receive 
$1 .00. If you solve fewer problems than the goal, you must pay back SI .00. 

You will not be told what the goal is, but what do you guess that it is? 

(Specify number of problems.) Work as hard as you can on this test. 
Based on your previous experience, knowing that Test 2 is equal in diffi- 
culty to the previous tests, please answer the following questions. 
How many problems do you personally hope to complete on 

Test 2? 

Do you believe that you should be penalized if you do not com- 
plete as many problems as you hope to? 

(yes or no) 
If your answer to the last question was "no," at w^hat level do you think 
we should set the goal for your performance? In other words, how many 
problems do you think we should require you to complete in order to receive 
the reward? 

Exhibit 4A.7. Sample budget — aspiration level combination 

(Group Ay) 



104 



Name 

QUESTIONNAIRE 

We would like to find out some of your reactions to the experiment. Please 
feel free to answer these questions candidly, and as fully as you like. 
Did you enjoy participating in the experiment? Why or why not? 



Did you feel a sense of competition with the other participants, or were you 
really competing only against yourself? 

Did the money rewards make you try harder? 

Would you have worked harder if the money rewards were higher — say $2.00 

per test throughout? 

Do you think the goals were: (a) too low; (b) just about right; or (c) too high 

for judging your performance? 

(a, b, or c) 
Were you satisfied with your performance on the experiment? 



Did you feel that you were working toward a goal that you had set up for yourself 
or one that we had set for you? Explain: 



Did you feel tense at any time during the experiment? 
How do you think the budgets* were set? 



Write down any comments you wish about the conduct of the experiment, or 
your own attitudes toward it. 



*"Goals" should have been used instead of "budgets" in this question. 
Exhibit 4A.8. Sample questionnaire 



105 



QUESTIONNAIRE PAGE 2 

How did you interpret the question, "How many problems do you personally 
hope to complete?" 

Did you find that there was much difference between that question and the one 
which asked about the number of problems we should require you to 
complete? 



Exhibit 4A.8 (conl.). Sample questionnaire 

Groups jS and 7 only 

TABLE 4a. 2. CLASSIFICATION OF SUBJECTS IN EXPERIMENTAL GROUPING 





Industrial 


Industrial 






Aggregate 




Management 


Administration 


Engineering 


"Years of Study" 




Soph. 


Jr. 


Sr. 


1st yr. 


2nd yr. 


Sr. 


Adv. Grad. 




Aa 


5 




1 




3 






32 


A0 


1 


1 




4 




3 




37 


At 




2 


3 


4 








38 


B« 


4 




1 




4 






36 


B/3 








5 




4 




41 


Bt 




2 


3 


3 






1 


39 


C« 


4 








4 




1 


38 


C/3 


1 






4 




4 




38 


C7 




2 


2 


3 




2 




37 


D« 


4 








3 




2 


38 


D/3 


1 






5 




3 




39 


Dt 




3 


2 


3 






1 


38 


Totals 


20 


10 


12 


31 


14 


16 


5 





JIP!^ 





^ 


Oi 


CD 


t^ 


CD 


I— t 




1 '~ 


o 


'-t^ 


cr: 


,-H 


CD 


.2 
1 


CO 


(M 


CM 


CO 




H 










T— I 




lO 


00 


c^ 


CO 


lO 




CO 


00 


CD 


l> 


00 




'a, 
< 












CO 




t^ 


CO 


(M 


'tl 


CD 


^-^ 


lO 


lO 


-^ 


lO 


lO 


o 


G 












CM 






o 


CO 


o 


CO 


CD 


^ 


^ 


CO 


^ 


'^ 


lO 


-c 




























m 




r- 


00 


^ 


C^l 


,_! 


CO 


^ 


CO 


-^ 


lO 


00 


^^ s 














I— 1 














^— ' 




rt^ 


r>- 


05 


00 


00 


1 


(M 


^ 


co 


•^ 


'Tt^ 




f- 


















Oi 


r- 


CO 


CD 


LO 




" 


(M 


(M 


CO 


CO 


04 




'^ 


(M 


'^ 


^ 


00 


00 




S^ 


O 


lO 


o 




lO 


-fj 


(N 


(M 


ca 


CM 


o 


a; 


H 












fat 






















1 


:3 




O 


.—1 


T— 1 


lO 


t^ 


« 


CO 


00 


1> 


00 


lO 


00 
















c^ 




(M 


lO 


o 


t^ 


■^ 




iO 


lO 


-* 


»o 


CO 


00 


c 














o 














'■+3 




^ 


r^ 


Tt^ 


C^l 


I^ 


c3 


T^ 


-<tl 


CO 


CO 


CO 


'l^ 
















■q. 














M 
^ 




(M 


00 


O 


Tt^ 


-^ 


CO 


TtH 


CO 


lO 


CO 


CD 






























1—1 


















T)^ 


00 


Oi 


CD 


r^ 


1 


(M 


Tfl 


CO 


^ 


CO 


CD 


01 




O 


lO 


O 


^ 


C2 




'"' 


CO 


(M 


CO 


C^l 


o 




^ 














1 « 


CO 


T— 1 


to 


t^ 


en 




-f 


Ol 


CO 


t^ 


t^ 




(M 


(M 


CM 


C^l 


as 




^ 


C5 


CD 


lO 


^ 




CD 


ir^ 


lO 


CD 


OO 


00 


>: 












CM 
















c 
o 




O 


r- 


lO 


^ 


CD 


lO 


Tt< 


CO 


CO 


^ 


lO 


-M 














a; 














b£ 














T3 




lO 


(M 


Tt^ 


lO 


S. 


3 


Tt^ 


CO 


CO 


CO 


CO 


CO 


CQ 












'"' 


1 




^ 


00 


Oi 


o 


00 


1 


CO 


CO 


CO 


CM 


-t 


CO 


a 












'"' 






Oi 


00 


o 


't 


o 




(M 


CO 


(M 


CO 


r^i 


to 






o 


t^ 


CM 


CTi 


00 




'"* 


CO 


(M 


CO 


Ol 


^ 




-t-^ / 












c 


cc / 

H Z 
/l 


^ ^ 




,, — ^ 






.2 
1 




m| 


a 






'a. 
< 


/ ^ 


B 




1 


^ 





H 
o 

<^ 
o 

Q 
> ft 

o o 

m « 
«. ^ 
S H 





.^ 


O -H CD -H 


00 


^ 


-1 03 


'^ CM C^l T-i 


Oi 


<! a 


00 t^ 00 00 






H 




CO 






CD 00 CD CO 


CO 




CD 


CO C3 Ol CM 


00 






CM ^ CM CM 


00 






G2 lO t^ to 


CD 




lO 


-H C^l CO CO 


^ 








to 






a: CM 00 o 


Oi 




^ 


^ o o ^ 


CO 








Tt^ 






O ^ CO CD 


CO 




CO 


CM ^ CM CM 


00 








r^ 






t^ CO l:^ 00 


to 




CM 


CM O CO Ol 


c^ 






r-( T— 1 1— 1 .— 1 


"^ 






Ci Oi to Oi 


C^l 




1—1 


00 Ir- O^ 00 


to 

CO 




/ 
















/ 






H 




<; PQ O P 




/ 


T3 




1=1 3 


/ 


:3 




<1 pq 


/ 


PQ 







107 



TABLE 4b.3. performance. PROBABILITY OF SIGNIFICANT DIFFERENCE 
BETWEEN ASPIRATION LEVEL GROUPS 



Group 


& 7 


a 

13 


.981 1.000* 
.998 



*To 3 decimal places. 



TABLE 4b. 4. 



PERFORMANCE. PROBABILITY OF SIGNIFICANT DIFFERENCE 
BETWEEN BUDGET GROUPS 





B 


c 


D 


A 


1.000* 


.369 


.680 


B 




1.000* 


.998 


C 






.393 



*To 3 decimal places. 



TABLE 4b.5. 



PERFORMANCE. PROBABILITY OF SIGNIFICANT DIFFERENCE 
BETWEEN NINE-MAN GROUPS 



(Determined by two- tailed ^-test with 106 degrees of freedom) 



Total -^ 
Score 

i 




277 


254 


246 


246 


235 


221 


218 


Group -^ 


Da 


B^ 


Bt 


Aa 


Ca 


Ba 


D3 


i 
















316 


Dt 


* 


t 


t 


t 








302 


At 




* 


* 


* 








297 


Ct 




X 


* 


* 








294 


c^ 




X 


* 


*■ 








292 


A^ 




X 


* 


* 








277 


Da 










X 


* 




254 


B^ 














X 



Symbol f * x 

Significance Level .1% 1% 5% 



108 



TABLE 4b.6. component ANALYSIS OF VARIANCE PERFORMANCE 







Sum of 






Variance 






Due to: 


Squares 


DF 


Mean Square 


Ratio 




Main Effects: 














Linear t 


1010.542 


1 


1010.542 


193.702 


t 




Quadratic t 


230.480 


1 


230.480 


44.179 


t 




Cubic t 


314.850 


1 


314.850 


60.351 


t 




R = quartic+ t 


1.628 


2 


1.628 


.310 




Total / 


1557.500 


5 








a 


77.121 


2 


38.561 


77.391 


t 




b 


53.315 


3 


17.772 


3.406 


* 


Interactions (2-way) 














a X linear t 


6.291 


2 


3.146 


.603 






a X quadratic t 


7.251 


2 


3.626 


.695 






a X cubic t 


.458 


2 


.229 


.044 






R = a X quartic+ t 


8.324 


4 


2.801 


.536 




Total (ta) 


22.324 


10 








b X linear t 


8.731 


3 


2.910 


.557 






b X quadratic t 


3.451 


3 


1.150 


.220 






b X cubic t 


6.507 


3 


2.169 


.416 






R = b X quartic+ t 


3.039 


6 


.506 


.097 




Total (tb) 


21.728 


15 








(ab) 


100.323 


6 


16.721 


3.205 


t 


Interactions (3-way) 














(ab) X linear t 


23.807 


5 


4.761 


.911 






(ab) X quadratic t 


6.995 


5 


1.399 


.268 






(ab) X cubic t 


13.737 


5 


2.747 


.526 






R = (ab) X quartic+ t 


19.317 


10 


1.932 


.370 




Total (tab) 


63.850 


30 








Error 


3005.111 


570 


5.217 






Total 


4901.278 


047 









t = test R = remainder 

a = aspiration *denotes significance at 5% level 
6 = budget fdenotes significance at 1% level 



109 



CO 



§.2 

> 


3.569* 
.266 





1 


1 


CD :3 


GO <M 


CO 


CO 

10 




L-O 

CO 


1 


02 s 




CO 

CO 


CO 


oi 

'I" 




§.2 

1^ 


00 Dl 


10 

CO 


' 


' 


1 § 


lO GO 


1 

01 


s 




£ 

1 § 

^ cr 

CC CO 


10 

i-I CO 


CO 


lO 




CO 


0) 

c 2 

•r <^ 

> 


TtH 00 

!>. CO 


CO 


1 


i 


1 ^ 


00 CO 
!>. CD 


00 





1 


^1 


t^ 00 

10 00 

CO -*' 


16 


88 

d 




< 




(M CO 


CO 


CD 







Main Effects 
Aspiration 
Levels (a) 
Budgets (6) 


Interaction 
Aspirations X 
Budgets (ah) 


0) 

c 

s 


3 



,— , o 



110 






CO 


> 


1.051 
1.110 


§ 

r-l 


1 


1 




,-1 CO 


CO 
CO 

CO 
I— 1 


-r 
o 

00 


' 


s g 

02 m 


CO O 


1— 1 

00 

o6 


lO 

id 
CO 

00 


to 

CO 

CO 




0) 

il 
> 


O CO 
O 00 
00 l:^ 

CO 




1 


1 




lO CO 
l>^ CO 


CO 
CO 


lO 


1 




Oi 00 
00 t^ 


CO 
CO 
CO 


CO 


CO 
CO 

2 




i.o 

> 


,—1 lO 




' 


' 


'^ Oi 


t^ CO 
00 CO 
t^ 00 


LO 


o 

CO 


' 


a ^ 

d cr 
xn. CO 




o 


00 
lO 


CO 
lO 


CO 


Q 


(M CO 


CO 




o 


Q 3 


Main Effects 
Aspiration 
Levels (a) 
Budgets (6) 


Interaction 
Aspirations X 
Budgets (ah) 


CD 

G 

si 


o 



111 



TABLE 4b.8. analysis OF VARIANCE - 


— GOAL DISCREPANCY 


(at — pt-i) 


Due to 


Sum of Squares 


DF 


Mean Square 


Variance Ratio 


Main Effects 










Tests (t) 


57.611 


4 


14.403 


2.285 


Aspiration 










Levels (a) 


4.225 


1 


4.225 


.669 


Budgets (6) 


117.142 


3 


39.047 


6.194tt 


Interactions (2-way) 










Tests X 










Aspirations (ta) 


9.845 


4 


2.461 


.390 


Tests X 










Budgets (tb) 


42.455 


12 


3.538 


.561 


Aspirations X 










Budgets (ab) 


29.897 


3 


9.966 


1.581 


Interactions (3-way) 










Tests X 










Aspirations X 


649.623 


12 


54.135 


8.587tt 


Budgets (tab) 










Error 










(Remainder) 


2017.222 


320 


6.304 


— 


Total 


2326.375 


359 


— 


— 



tfdenotes significance at .1% level. 

TABLE 4b. 9. ANALYSIS OF VARIANCE ACHIEVEMENT DISCREPANCY {a t — p t) 



Due to 


Sum of Squares 


DF 


Mean Square 


Variance Ratio 


Main Effects: 










Tests (0 


591.128 


4 


147.782 


17.790tt 


Aspiration 










Levels (a) 


7.226 


1 


7.226 


.870 


Budgets (b) 


122.075 


3 


40.692 


4.S99t 


Interactions (2-way) 










Tests X 










Aspirations (ta) 


2.316 


4 


.579 


.070 


Tests X 










Budgets (tb) 


50.605 


12 


4.217 


.508 


Aspirations X 


50.674 


3 


16.891 


2.033 


Budgets (ab) 










Interactions (3-way) 










Tests X 










Aspirations X 


64.754 


12 


5.396 


.650 


Budgets (tab) 










Error 










(Remainder) 


2658.222 


320 


8.307 


— 


Total 


3520.864 


359 


— 


— 



fdenotes significance at 1% level, 
tfdenotes significance at .1% level. 



112 



CHAPTER 5 



A Mathematical Model for 
Budgetary Planning 



5.1. Introduction 

To this point in the thesis the emphasis — both experimental and 
analytical — has centered on individual performance. Qualifications 
were introduced from time to time in order to draw attention to other 
problems relevant to budgetary management. One such problem 
involves issues of coordination between departments, a problem of 
particular importance insofar as behavior in one department may affect 
(favorably or adversely) the conditions of operation in another. Where 
such relations — called "factor cooperancy"^ and "rivalry" in technical 

^The term "factor" of production should be emphasized, because the economic 
theory of production is based on a model of the firm which distinguishes only between 
an entrepreneur (at the top) who makes all "basic" decisions, and factors of produc- 
tion, (labor and capital) that follow to the result intended by the entrepreneur. In 
particular, this theory does not allow for intervening tiers of management — distin- 
guished as planning, operating, and control agents by W. W. Cooper (20) — and 
the kinds of immediate or ultimate consequences such intervention may have for any 
particular firm. It follows that there is no problem other than that of resource 
allocation (which occurs only along a technological optimum) and, in particular, there 
is no problem of control — e.g., aspiration levels of intermediaries, report content 
(or timing), etc. (Cf., e.g., H. Guetzkow and H. A. Simon (71) for an experimental 
study which shows that the realization of an optimum depends, at least in part, on the 
kinds of information, organization arrangements, etc., supplied to participants.) 
However, the ba ic categorizations such as efficiency (defined in alternative cost 
terms), marginalism (broadly conceived), cooperancy, rivalry, etc., do carry over as 
useful constructs at a more fundamental level, and the current economic model of the 
firm should therefore not be allowed to stand in the way of the use of these concepts 
merely because of inadequacies in the vehicle used for their transmission. 

113 



114 A Mathematical Model for Budgetary Planning 

economics — are present, it then follows, as is well known from economic 
theory, that optimization undertaken department by department does 
not guarantee an overall optimum. ^ 

It is clear that pursuit of the techniques advanced in earlier chapters 
could at best produce rules for obtaining the optimum in each depart- 
ment taken separately. A simple example will suffice to show that the 
definition of an optimum for a single department cannot be made in 
vacuo. Recognizing the rather obvious interdependency of cost and rate 
of production, should the decision be to increase production at an 
optimum rate in a given department, to keep unit cost constant, or to 
reduce cost at an optimum rate while keeping production constant? 
This question cannot, of course, be answered with the aid of the control 
mechanisms discussed earlier. Expressed in other terms, the perfection of 
techniques for controlling the unicellular organism is a futile undertaking 
(from the point of view of budget control if not that of psychology) unless 
some means of establishing subgoals for each of the controlled units based 
on the overall aims of the entity can be developed. 

It is the purpose of this chapter to explore this topic in some depth. 
This will be done by means of the techniques of linear programming 
which appear to be the most suitable of the available anah^tic methods 
for this purpose. A particular model will be synthesized which has the 
following advantages: (a) it exhibits the desired features in a simple and 
straightforward manner; and (b), because of its special structure, admits 
the possibiUty of evolving especially efficient methods of solution. ^ 

Although the study to be conducted in this chapter has its most 
obvious relevance in the area of planning,^ it also has rather clear 



^An overall optimum would then be secured only in the case of "independence" 
(as distinguished from rivalry and cooperancy). Cf. Sune Carlson (5). 

Incidentally there are further possibilities that require attention such as "optima 
in the small" and "optima in the large." These need not be dealt with here, however, 
since linear programming, when applicable, guarantees the emergence of the latter. 
See, e.g., A. Charnes, W. W. Cooper, and A. Henderson (15). 

Wide, A. Charnes and W. W. Cooper, "Management Models and Industrial 
Applications of Linear Programming," Management Science (10), where the import 
of such special model types is discussed for its future bearing on (a) appUcation and 
(b) scientific research. 

^In the sense of choosing between alternatives when, as is generally true in 
economics, no issue of follow-up or implementation is involved so that no problem 
of control is involved. Examples of this kind are the pricing delegation studies to be 
found in T. C. Koopmans (45c) or A. Charnes and W. W. Cooper (14). The latter, 
it should be noted, does discuss the possibility of control problems emerging (e.g., 
in their discussion of contradictory systems), and the point is even more strongh^ 



A Mathematical Model for Budgetary Planning 115 

implications for the kind of budgetary control problems which are of 
interest for this thesis. Thus, as indicated above, a crucial issue which 
may emerge is the specification of rules for guiding or stimulating 
departmental behavior in certain directions in order to achieve certain 
overall results. For instance, a question may be asked whether costs 
(including transfer costs and prices) should be reported accurately, and 
be based on fair and equitable procedures. Or is it better — e.g., in 
achieving an overall optimum — to adopt an alternate approach and 
report costs to each supervisor which, though distorted from the stand- 
point of the overall entity, nevertheless succeed in furthering entity 
objectives? Finally, if either approach be adopted, how fair (or unfair) 
or how accurate (or inaccurate) should the reports be? 

While these topics cannot be fully explored here, and while exper- 
imental (or empirical) evidence of a kind that might be desired is not yet 
available, there is some advantage in conducting preliminary analytic 
explorations. Minimally, it should then be possible, at least in principle, 
to explore some implications of various costing, reporting, and budgeting 
rules that are of interest because of their potential relevance for currently 
employed methods in practice. 

Such an exploration, then, constitutes the main objective of this 
chapter. The task will be regarded as complete when main issues have 
been clarified, and no effort will be made (as in the preceding chapters) 
to carry things forward to a resolution which, in the end, must depend 
on some measure of empirical verification. In the process of clarification 
an effort will be made to provide details for synthesizing certain kinds 
of models and to supply prescriptions for a procedure of solution that 
will, at least in some situations, provide a beginning for applying the 
results of the inquiry which is to be conducted. 

5.2. The Hierarchy of the Factors of Production 

The management science literature has dealt largely with problems 
in which the strategic factors^ are a priori determined. For example, in 
the machine loading problem the amount of each output is determined 
solely from the costs and limitations associated with the set of factors 



hinted at by Charnes and Cooper (10) when the possible desirability of supplying 
"misinformation" to subordinates is noted. Also of interest in this connection are 
studies which deal with decision-making by teams, such as by J. Marschak (58) 
and R. Radner (63a). 

^For a discussion of the specific meaning of strategic factor as it is used here, 
see Barnard (3). 



116 A Mathematical Model for Budgetary Planning 

"machine hours. "^ An implicit assumption in a model of this type is 
that the costs of other factors of production are either negligible or are 
proportional to the amount of machine-hour usage or output. Further- 
more, it is necessary that availability restrictions on other factors are 
redundant — i.e., that the set of things known as "machine hours" is a 
strategic factor. To put the matter another way, the felicity of any such 
model depends vitally on whether or not it incorporates all of the relevant 
bottlenecks relative to the criteria which are of interest as stated (or as 
they should be stated) in the objective function. 

In order to carry this analysis a step further, assume that there is a 
machine manager who administers the operation of the several machines. 
From his vantage point, when he "solves" the machine loading problem 
(i.e., maximizes profit or minimizes cost subject to certain output 
requirements and restrictions) he will discover, within the strategic fac- 
tors, limits on the capacity of one or more machines (or their operators). 

If the assumption that "machine hours" is the strategic factor is 
removed, it is fairly easy to visualize a labor manager and a materials 
manager on the same level of the hierarchy as the machine manager. ^ 
As viewed from the vantage point of the department head to whom these 
three report, the operations of any or all of the machine managers may 
be looked upon as a strategic factor. Proceeding an additional level up 
the organizational ladder, the supervisor to whom this department head 
and several others report can either view each department as a single 
factor; or he can investigate a level deeper, considering as factors labor, 
materials, machine hours (and even managers at much lower tiers) in 
each department. 

In the discussion that follows, it will be assumed that each super^dsor 
can "see" two levels below him. In this sense an attempt will be made 
to focus on features of what Charnes and Cooper (10) call a "hierarchial 
model." 

5.3. The Concept of Limited Substitution — A Definition 

The production functions of economics have certain drawbacks 



^See, for example, Cooper and Charnes (10), pp. 48-57; and for another example 
in the scheduling of an oil refinery, Manne (56) deals with crude oil availabihty as 
a limiting factor. 

2It is not actually necessary that there be a "machine-hour manager" as such; 
it is equally possible to consider these managers as "chunks" of supervisory time, 
time in which the department head spends in managing these factors, as opposed to 
time spent in supervising the managers. Clearly, at some level in an organization 
a point will be reached where a man is managing things rather than people. It is 
expected that this will be true at the lowest level (i.e., production workers), partially 
true at the next higher level, etc. 



A Mathematical Model for Budgetary Planning 117 

including lack of empirical validation on the one hand and of easy access 
to managerial data (or even ways of thinking) on the other hand. This 
suggests that the topic at issue may best be formulated in some other 
manner, if possible, since there is little advantage to piling one difficulty 
upon another when empirical validation is ultimately undertaken. The 
approach (e.g., via linear programming) that has been common in 
management science has a certain appeal. But as pointed out above, 
the efforts here have (to date) been restricted to isolating one set of 
strategic factors and to concentrating only upon the factors thus isolated 
in a way which may best be described as focusing on one level of manage- 
ment. Such an approach has limitations which become apparent in cases 
such as those where an a priori choice of strategic factors is not obvious. 
Of course, if the problem is sufficiently simple the limitational factors 
may be verified, perhaps easily, in one form or another by direct obser- 
vation; in larger problems parameterization techniques may be used to 
secure required guidance. But in many situations it may not be possible 
to depend on either of these modes of validation, and other possibilities 
may then need to be considered. In particular, it may become necessary 
to consider the synthesis of procedures which will raise alarms or danger 
signals when one strategic factor or another has been omitted from the 
model (explicit or implicit) used to plan the activities that will be 
undertaken. Another case of interest that arises is one which will be 
called ''limited substitution.'^ Here the use of mutually exclusive 
categories, which is assumed, may be misleading, especially in a linear 
model; nevertheless, it is assumed that, under certain conditions, it is 
possible to effect substitutions (or at least limited substitutions) within 
the original groupings. ^ 

5.4. Form of the Production Function Under Limited Substitu- 
tion2 

It is readily seen that the assumptions of limited substitution lead 
immediately to a production function in which the output of each item 



iThese involve issues of aggregation and model detail which cannot be discussed 
here at any length, aside from the obvious comments already entered. That the issue 
of "linearity" is involved can be seen from the fact that the indicated substitutions — 
at some specified level of aggregation — would make their effects felt via the coeffi- 
cients (which then become functions of the outputs and mixes). 

2The limited substitution model presented here was developed independently by 
the author and reported in (76). I was given access to the unpublished paper by 
Charnes, Cooper, and Miller (18) after the formulation was completed. The model, 
although a special case of their general dyadic formulation, is not specifically described 
in that paper, and, in any case, the computation scheme stands as an original develop- 
ment in its own right. 



118 A Mathematical Model for Budgetary Planning 

is limited to the amount (in terms of product-equivalents) of the factors 
that will be allocated to its production within each factor subgroup. 
Under optimization the choice is designated to the factor subgroup which 
''costs" the least. Determinacy, on the other hand, requires that at least 
one such factor subgroup be strategic, since otherwise no limit to 
expansion would be present. 

In addition to assuming that at least one strategic subgroup exists, 
it is further assumed that the substitution within groups is linear. The 
limitation on the output of any item may then be expressed as a set of 
linear inequaUties where the number of inequations is equal to the 
number of factor subgroups from which inputs are required for this 
particular item. 

For the model to be considered, let there be p factor subgroups, and 
let it be supposed that the ith item requires inputs from Qi of the sub- 
groups. Further, let 

Xijk = number of units of the A;th factor of the jib. subgroup used 
in production of the ith. product ^ 

(5.4.1) aijk = number of product equivalents of the zth item produced 

per unit of x'ijk 

yi = amount of the ^th item produced 
Uj = number of factors in the jth subgroup 
The condition to be imposed on the iih. item may then be written^ 

nj 

(5.4.2) 2/i<Z) 2 O'ijkx'ijk i=lj...,m 

jeqi k =1 

iThe reason for the use of primes on x and c will become evident in equations 5.4.7 
where the transformation performed on these variables is shown. 

m only a subset of the n,- factors is used in producing the ith. item, then for those 
nonutilized factors, it is assumed that 

aijk = 1 

c'ijk = M 

where, as usual, M is a large penalty associated with an artificial process whose cost 
is therefore so high that it will not appear at an optimum when solutions exist. See 
(10), p. 50. Where nonutilized subsets are known in advance they ma}', of course, 
be eliminated from consideration at the outset. The device suggested here is intended 
for ex post facto adjustments where, after an initial optimum has been achieved, 
it is found desirable to "derive" some of the factors out of positive use. 



A Mathematical Model for Budgetary Planning 119 

Noting that the factor subgroups are mutually exclusive, the factor 
availabilities may be written : 



(5.4.3) E <ik < h 



i=i 



jk 



where hjk is the maximum amount available of the jth subgroup in the 
kih. factor grouping. This restriction states that the total amount of 
factors, considered over all products, cannot exceed a prescribed number, 
hjk for A: = 1, ...., nj\j = 1, ...., p, so that 

p m p 

(5.4.3a) 2 .X) ^ijk < 2 ^jk = hk 

and 

nj m nj 

(5.4.3b) i: X; x-jk< E hjk = bj 

k=l i=l k=l 

In other words neither the amount, hk, of any factor group or the amount, 
bj, of any subgroup can be exceeded either in any or all of the activities 
to be undertaken. 

The problem is to maximize the difference between the contribution 
of the outputs and the costs of the necessary inputs. 

Let 

d = the contribution per unit of the iih item to profit and 
/5 4 4\ overhead 

c'ijk = cost of one unit of Xijk 
An analytical expression of the function to be maximized is therefore 

TO nj 

(5.4.5) ^ {dyi - X) £ c'iik x'ijk) 

i =1 jeqi A; =1 

and the maximization is to be undertaken over the previously stated 
constraints and is also to take into consideration restrictions of the form 
(which may emanate from inventory and sales considerations) 

(5.4.6) U < Vi < Ui 

This last restriction is in what is called bounded variables form which, 
as Charnes and Lemke (16) note, is typical of many management 
problems involving ''balance restrictions."^ 



iThus the Li and C/» may be, say, lower and upper limits to inventory, or else they 
may emanate from the need for supplying certain minimal (and not more than 
maximal) amounts to certain markets. 



120 A Mathematical Model for Budgetary Planning 

It is now assumed — although this is not essential — that the number 
of factors exceeds the number of products. For convenience it is now also 
desirable to introduce certain transformations which ''normahze" or 
express the production restrictions in terms of ''unit" amounts. For 
this purpose, let 

(5.4.7) bij, = ^ 

Clijk 
Cijk ^^ 

aijk 

Using (5.4.7), factor usage can be expressed in units of product equiv- 
alents of the outputs for which they are used. If the djkS and aijkS are 
positive and no lower bound (other than those emanating from the 
constraints and non-negativity) is placed on the Xijk's, the inequality in 
(5.4.2) can, for purposes of optimization, be replaced by equality, since 
it will never be most profitable to utilize a combination of factors which 
will satisfy any of these constraints as a strict inequality. The problem 
may thus state in the following final form 



Maximize: 



Subject to 



i=l jeqi k=l 



2 ^ijk — 2/i = e = 1, ..., m;jeqi 



m 

(5.4.8) J2 bijkXijk <bjk k = 1, ., Uj; jeqi 

Vi < Ui i = 1, ...,m 

-Vi < -Li 
The dual to the above problem is^ 
Minimize : 

p ny m m 



iFollowing the notation for the dual variables used in Danzig's row-column sum 
method of solution of the transportation model described in (45a). 



A Mathematical Model for Budgetary Planning 121 

Subject to 

Rij + hijkKjk > —Cijk I = 1, ..., m; jeqi 

(5.4.9) k= 1, ...,nj 
— 2 Rij + Ti - I3i > Ci i = 1, ...,m 

Jeqi 

Kjk, Ti, ^i > and Rij is unrestricted in sign. 

The theorem of the alternative in hnear program asserts that strict 
inequality cannot, at an optimum, simultaneously obtain for (1) a dual 
variable and the corresponding direct constraint or (2) a direct variable 
and its associated dual constraint. ^ 

From (1) it is clear that, at optimum, Ti > implies yi = Ui and 
iSi > implies yi = Li^ Furthermore, yi < Ui implies Ti = and 
Li < yi imphes ^i = 0. Therefore, utiUzing (2) and assuming the 
nonexistence of degeneracy^ the conditions for optimality are 

Rij + bijk Kjk = —Cijk Xijk > 
>: —Cijk elsewhere 
2 Rij = Ci Li < yi> Ui 

(5.4.10) - E R^j < Ci yi = Ui 

Jeqi 

— 2 Rij > Ci Vi = Li 

jeqi 

5.5. Example of a Limited Substitution Problem 

For exposition purposes a model has been chosen which utilizes three 
factor groups to be called ^'labor," "materials," and ''machine hours." 
Three products will be produced. Products 1 and 2 require inputs from 
all three factor groups, but Product 3 is a hand operation, requiring labor 
and materials only. There are three types of labor, two types of material, 
and four types of machines. 

The restrictions of the model are presented in simplex form in Figure 
5.1 in order to bring out the details of model structure by reference to a 
standard format used in this field of analysis. Structurally (i.e., without 
regard to numerical magnitudes of the coefficients) the resemblance to a 
transportation model is striking. If, in fact, complete substitution 



Wide (81). 

2For a more complete treatment of the bounded variables problem, see (13) or (16). 

Wide (9) for methods of resolving degeneracy. 



122 



A Mathematical Model for Budgetary Planning 



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A Mathematical Model for Budgetary Planning 123 

between groups existed, the tableau could be compressed into a trans- 
portation schema by suitably arranged scalings. However, a special 
(and more efficient) computational scheme may be synthesized from 
the prescriptions for ' 'sub-dual algorithms" as set forth in Charnes, 
Cooper, and Miller (18). ^ 

Labor yi = ixju + ix[i2 + i^ns 



(5.5.1) 



Material 


2/1 = K21 +K22 


Machine Hrs. 


yi = 1.2a:'i3i + lx\^^ + lx[^^ -f .8a:'i3 


Labor 


2/2 = 2^211 I 2^212 + 2^2 13 


Material 


2/2 = i42 1 + 3^222 


Machine Hrs. 


?/2 = 1.8x'23i + 1.2x'232 + Ix^gj + M 


Labor 


2/3 = TVx'312 +ia:'3i3 


Material 


2/3 = la:'321 + 1^22 



It is seen that a unit of Product 1 may be produced using 5 units of 
the first type of labor, 4 of the second type, or 3 of the third. But a unit 
of product also requires either 5 units of the first type of material or 4 
units of the second, and similarly requires 1/1.2 units of machine type 1 
time, 1 hour of machine type 2, 1 of type 3, or 1/.8 of type 4; similarly for 
products 2 and 3. It should be noted that Product 3 is "short-changed." 
It requires no machine time, but also cannot use labor type 1 which (as 
shown below) is the cheapest labor source. It might be surmised that 
Product 3 is produced by a reasonably-skilled hand operation which 
requires labor which is more skilled than type 1 and is seen to be more 
"sensitive" to the increase in skill between labor types 2 and 3 than 
either of Products 1 and 2. 

It should be re-emphasized here that the production restrictions 
might have been stated in the form of inequalities — e.g., yi requires 
at least so much labor — but as the CijkS that will be used are positive, 
and no lower bounds are placed on factor usage, ^ no slack would exist 



lit may be noted, however, that the instructions supplied in this article are of an 
extremely general character and will probably remain so until, as these authors note, 
an "algorithm for generating algorithms [including requisite models and transfor- 
mations] have been completely prescribed." In short, it is by no means certain in 
which direction to proceed from these general instructions. 

2It is reasonable to believe that for "worker morale" or other long-term considera- 
tion, lower bounds might be placed on labor. This can be arranged by treating labor 
slack as a bounded variable, but is avoided here to prevent increasing the complexity 
of the exposition. 



124 



A Mathematical Model for Budgetary Planning 



at optimum; therefore, as pre\dou5ly noted, an equivalent equality may 
be used instead. 

The product contributions per unit of output and factor costs per unit 
of input are sho^n in Figure 5.2. 



Factor Group 


Labor 


Material 


Machine Hours | 


Factor Type 


1 


2 


3 


1 


2 


1 


o 


3 


4 


Symbol 


c'.ii 


c'iX2 


c\.^ 


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c'iSi 


c'.-3 2 


c'iii 


c'iZi 


Unit Cost 'Si 


1.8 


2 


2.5 


4 


3 


7.2 


6 


5 4 i 



Product 


1 1 2 


3 


Symbol 


Ci 


Ci 


Cz 

25 


Contribution 


30 


20 



Fig. 5.2 

The product requirements are 

50 < ?/i < 100 
(5.5,2) 100 < ?/2 < UO 

SO < ?/3 < 120 

and the factor availabihties are 



(5.5.3) 



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A Mathematical Model for Budgetary Planning 



127 



Applying the linear transformation shown in (5.4.7) to the Xijk^ and 
adding the artificial variable Xan with a cost of %M, the problem may be 
comprehended in the tableau of Figure 5.3. 

5.6. Computation Scheme Format 

For computation purposes the variables are arranged in the form 
which is called "dyadic, "^ as shown in Figure 5.4. Typical cell detail is 
given in Figure 5.5. 



1 Cijk 




1 h,.. 



u. 1 


1 c, 




----'^ 


u - 



Input 
(a) 



Output 
(b) 



Fig. 5.5 



The purpose of the hi entry in Figure 5.5(b) is to allow an overall 
restriction on production 



(5.6.1) 



E hiViKB 



to be placed on total production to take into account a total warehouse 
restriction, if desired. In the example, it is assumed that hi = 1, for all i. 
5 is a bound, sufficiently large so that it is greater than any possible S?/i 
(i.e., the restrictions associated with 5 are "dummies" whose function is 
to preserve the rectangular form of the tableau). 

5.7. Optimization Procedure 

The computational details that have been devised may now be set 
forth as follows. 



Initial Solution, The first step in finding an initial solution is to 
set yi = Li for the products, underlining the entries to note that these 
bounded variables are equal to their lower bounds. ^ The dummy 
restriction is satisfied by entering B = '2yi in the last column of the last 
row. 



^Cf. Charnes, Cooper, and Miller (18). 
2See (10) or (16). 



128 A Mathematical Model for Budgetary Planning 

The filling of the Xijk entries follows the Northwest Comer Rule^ of 
transportation type models, considering each of the factor groups as a 
subproblem in the transportation model, with the yi column as the 
requirements column. ^ The machine-hour subproblem is considered as 
a reduced problem, satisfying only requirements 1 and 2, and hence 
having three rows (including slack) and four columns. 

It will be noted from Figure 5.3 that there are seventeen equations 
(not including the bounding equations on yi). The dummy restriction 
will add another, thus making the number of basis elements equal to 
eighteen. The filling-in process which has been used provides m -\- n — 1 
entries^ in each of the subproblems, as required for a basis, plus one 
{B — Zyi). Thus the labor group provides 4 + 3—1 = 6 elements; 
material provides 4 + 2 — 1 = 5 elements; and machine hours, 3 + 4 — 
1 = 6 and B — Xi, 1 or a total of eighteen active basis elements. A 
glance at Figure 5.3 will reveal that, provided none of the ?/,'s are in the 
basis {Li < yi < Ui), any two vectors which represent the coefficients 
for factors of different groups must be linearly independent. The method 
used for filling the squares in the subproblems assures linear independ- 
ence of the vectors within the factor group, and the number of cells 
utilized therefore guarantees that an active basis is achieved. The rules, 
then, for handling the bounded variables guarantee that the entire space 
in which the problem is exhibited will be spanned. 

Computation of the border entries. Following the method of 
Dantzig,^ as expanded and generalized by Charnes and Cooper (18), 
the RikS and KjkS are determined as follows 

Rule (i) Set Rik + bijkKjk = Crjk 

for all filled cells. Also 

(5.7.1) (ia) Set Rik = Kjk = 

for any cell in which positive slack appears, since then the theorem of 



^For discussion of the "Northwest Corner Rule" and basis requirements for the 
transportation model, see (13). 

2This process will cause no difficulty unless slack is a bounded variable. Should 
the case occur in which a slack variable must exceed its upper bound in order for 
yi = Li, the slack should be treated as an artificial vector with a lower bound equal 
to its original upper bound. As soon as the slack is reduced to its original upper bound, 
the slack variable can take on its original limits. 

Uhid. 

4(45a). 



A Mathematical Model for Budgetary Planning 129 

the alternative applies so that (when an optimum is attained) there will 
be the dual variables which are wanted. 

With these conditions satisfied the test for optimality may be divided 
into two parts: 

Rule (ii) Rik + bijkKjk > Cijk 

for all unoccupied factor squares. When all of these conditions are 
satisfied, no further substitution within a factor group can be profitable. 
This is a necessary condition for optimality, as may be proved mathe- 
matically; or following an easier course, it may be justified by economic 
consideration. A further necessary condition is 

Rule (iia) — ^ Rik > Ci for all yi = L, 

k 

— ^ Rik < Ci for all yi = Ui 

which will indicate that no further changes in production will be profit- 
able.^ These rules considered collectively are then seen to be both a 
necessary and sufficient condition for an optimum. ^ 

Iteration procedure. Although it is not clear whether first pro- 
ceeding with the substitution within groups will lead to an optimum in 
fewer iterations than first adjusting the production requirements, an 
apparent advantage of beginning with intra-group substitution is that, 
as long as yi = Li, the factor groups may be treated as separate trans- 
portation models in which iteration is relatively simple.^ 

The suboptimization within the factor groups proceeds exactly in the 
same fashion as that of the standard transportation model, hence will not 
be discussed.^ 

Figure 5.6 shows the tableau after the suboptimization has been 
performed, and the restrictions of Rule (ii) have been satisfied. Applying 
Rule (iia) to the revised tableau, it is noted that additional profit may 
be made by increasing the production of any of the three products. 

Let 2/1 be brought into the basis. The computational rule to effect the 



^Since the dummy restriction was stated such that slack will always exist, it follows 
from the theorem of the alternative, that the corresponding dual variable will always 
be 0, and hence need not be considered. 

2A proof, if desired, may be secured by reference to the Gale-Kuhn-Tucker dual 
theorem (45b). 

^In fact, if production is stipulated exactly — i.e., Li = yi = Ui, the problem is 
trivial — and involves nothing more than the separate optimization of the sub- 
problems. 

^Vide (10), (13), and (45a). 



130 



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132 A Mathematical Model for Budgetary Planning 

adjustments necessary for this are an adaptation of the stepping stone 
tours of Charnes and Cooper (13) which ''surround" the cell into which ?/, 
is to be brought with alternate +'s and — 's, utihzing the yi column as an 
addend to the factor group tableau but reversing + and — signs in this 
column. In order to choose the vector to be eliminated it is necessary 
to choose the minimum of the ( — ) entries, adjusted by their coefficients. 
(If all hijk = 1 the least of the ( — ) entries numerically would be chosen.) 
In this case the limiting entry turns out to be X233. The tableau is 
amended as shown in Figure 5.7 with yi no longer equal to its lower 
bound, and hence in the basis; whereas, while the (-|-) entries in the 
surrounding squares are increased, the ( — ) entries decreased. The border 
entries for the machine-hour group must be changed, with the restriction 

(5.7.2) -E^ifc^ci 

k 

substituted for the restriction 

(5.7.3) R23 + h2zKz2 = C233. 

Examination of the third tableau indicates that product 3 appears to 
have a much greater unit profit potential ($6.00) than product 2 (S2.40), 
so it will be brought into the basis. The product 3 square is surrounded 
as shown by the + 's and — 's in Figure 5.7. The limiting cell is slack labor 
type 2. The +'s are increased and the — 's decreased in both labor and 
materials. (Since no machine hours are required for product 3, these 
entries are unaffected, and since changing yi would require an additional 
basis vector in machine hours, paths including yi are not feasible.) 

The fourth tableau reveals that Xm can profitably be brought into 
the basis. Place + and — signs in the paths which utilize alternating 
horizontal and vertical moves, including the yi column, except that 
the signs are reversed in that column.^ The minimum value is clearly 
xm = 20. However, an increase in ys will take place. Should this 



^The paths would appear to be 

+ - - + + 

Xns -^ Xzi3 — > ^3 — * 5 — » *Sii 
and 

- + + 

JDut the latter path would require an additional basis vector in machine hours similar 
to the situation in the third tableau. It will be noted that the introduction of one of 
the Tji's into one basis requires the removal from the basis of a vector in one of the 
factor subgroups. A reduction in that yi would require an increase in one of the other 
yi'a thereafter — impossible in this instance since ys is not restricted by machine 
hours. 



A Mathematical Model for Budgetary Planning 



133 



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136 A Mathematical Model for Budgetary Planning 

increase have caused another variable — say slack in material 2 — to 
become negative, then this variable should have been removed from the 
basis instead and the other entries adjusted accordingly. The fifth 
tableau is shown in Figure 5.9. 

Profit can still be gained by introducing additional product 2. This is 
done, eliminating slack in material 2, and the obtained optimum tableau 
is shown in Figure 5.10. 

The value of the program at each stage may be computed either from 

(5.7.4) ^ = 2 CiVi - SZ)Z1 CijkXijk 

i i j k 

or, if desired, by^ 

i k j k 

The gain at each stage of the tableau may be computed in simplex 
fashion. If the increase of the variable above or its lower bound (or 
decrease from its upper bound) is denoted by Axijk or Az/i, as apphcable, 
the increase is 

(5.7.5) Az = —AxijkiRik + hijkKjk — Cijk) 

or 

Az = Ay^ia + 2 Rik) 

5.8. Power of the Model 

Examination of the sixth tableau reveals that the criterion presented 
to the department head by his supervisor — profit maximization subject 
to output limitations — can be translated into operational instructions 
to the managers of the factor subgroups. For example, the labor manager 
can be instructed to: 

1. Utilize all of labor type 3 in the manufacture of product 3. 

2. Utilize all of labor type 2 in the manufacture of product 1. 



^The first term of the summation will be for Li < yi < Ui. The functional 

of the dual is 

2 hjkKjk + TiUi - ^iLi 
j,k 

When yi = Ui, the dual restriction must be satisfied exactly, or 

Ci = -2i2i;t + ri 
k 
Similarly, when yi = Li 

Ci = -i: Rik - /3i 
k 



A Mathematical Model for Budgetary Planning 



137 



3. Utilize labor type 1 on product 1 up to the point where the 
combined efforts of labor types 1 and 2 reach 70 units of product 1 ; 
utilize part of the remaining type 1 labor to produce 115 units of 
product 2; send the remaining type 1 labor home (or to the 
company labor pool). 
Alternatively stated, the problem can now be handled by sub- 
optimization within the factor groups, taking the output stipulations 
found in the larger problem as the requirements. The control problem 
is now merely one of enforcing adherence to the production coefficients; 
i.e., the 600 hours of labor type 3 available must produce 100 units of 
product 3, etc. The intra-group relative seriousness of inefficiency in the 
various labor types can be comprehended in the dyadic format model 
tableau of Figure 5.11. It will be noted that, taking product require- 



\^ Labor 












\. Type 


1 


2 


3 


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^i 


Product X^^^^ 
















-9 




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100 


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4 


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Labor Sub-optimization 
Fig. 5.11, Transportation model 



138 A Mathematical Model for Budgetary Planning 

ments as given, the waste of 1 hour of type 3 labor costs S3. 00. It is 
essentially the cost of an additional unit of product 3 (S18.00) di\dded by 
the number of hours required per unit of product (6). Alternatively, it is 
the cost of introducing a unit of slack at the hourly pay rate of S2.50, 
with the resulting Ri + Kj — dj = $3.00. A similar waste of type 2 
labor would result in an additional cost of $2.25 while a waste of type 1 
labor would result only in a loss of $1.80, the hourly cost of the labor. 
Conversely, an improvement in the coefficients will produce cost savings 
of an equivalent amount. Therefore, the department head should set 
higher goals for the labor manager for efficiency of type 3 labor, relative 
to expected performance, than for type 2 and higher for type 2 than for 
type 1 (avoiding the error of setting the goals so high that they lose 
reality for the labor manager). Another possibility is to leave the intra- 
group optimization to the labor manager entirely, in which case the latter 
would set the goals for the various types of labor within his jurisdiction 
as indicated. 

The model also provides the department head with information 
concerning capacity evaluations. Clearly each capacity whose Kjk has 
a positive value is a strategic factor. Is it possible that the 600 available 
hours of type 3 labor is an item which is established by ''tradition?" 
Would it be possible to acquire this labor at a premium not to exceed 
$1.33 per hour? The model directs the department head's attention to 
the possibility of obtaining one type of additional labor or another, as 
opposed to the more nebulous form in which this question is often asked ; 
i.e., ''Should we or should we not hire more workers?" Furthermore, 
he can compute the gain from hiring additional labor. Hiring 120 more 
hours of labor type 3 would enable him to increase his production of 
product 3 to the upper limit of 120 units with a resulting increase in profit 
of $160 less any premium he would need to pay above the standard rate 
of $2.50 per hour. 

An interesting contrast between this model and "management by 
exception" develops in the context of paying a premium for a factor in 
order to increase profit. From the mathematical properties of the 
optimal solution to the linear programming problem, the increase in 
the amount of any factor for which Kjk > will result in increased 
profit, even if a premium (less than Kjk) must be paid on the additional 
units. Assume that the dj's are both standard and actual unit cost. 
A supervisor who chose to pay the premium to increase the "profit" of his 
department would, under a standard cost system, incur an unfavorable 
cost variance. Hence, under a usual scheme of "management by excep- 
tion," he would at least be required to file a report to show cause but 



A Mathematical Model for Budgetary Planning 139 

would not receive compensatory reward. In other words he would not 
be motivated to increase profit at the sacrifice of one of his ''black" 
variances. Thus the paramaterization has pointed up an apparent flaw 
in the principle of "management by exception." In its concentration on 
unfavorable variances, it creates a situation in which it is not desirable 
to increase one cost by some amount in order to increase profit by a 
larger amount. 

The complete absence of slack in the material subgroup has additional 
implications. First of all, the absence of slack in any factor group which 
contains required inputs for all the outputs (or a set of subgroups without 
slack which, among them, contain inputs for all of the outputs) renders 
the set of optimum outputs a Pareto point. Hence, the material sub- 
group in the example and, generally speaking, any slackless factor group 
taken as a whole is a strategic factor. If, in response to his supervisor's 
orders, the department head is required to increase the output of one of 
the products, he must choose either decreasing the output of one of the 
others or finding some way of altering the restrictions of the slackless 
group (s). 

5.9. Parametric Programming of the ModeP 

Changing the contributions of the outputs does not indicate, at least 
in the example, great sensitivity of the production schedule to these 
''intra-company trading prices." Although the optimum shown is valid 
only for $29.20 < c\ < $30.20, yi is increased to 81 J, yi decreased to its 
lower hmit of 100 for $30.20 < Ci < $43.00. Between $43.00 and $50.00, 
yi increases another 4-^^ units with a 17-f- unit decrease in ?/3, while a 
ci > $50.00 is needed to drive y^ to its lower limit. No change in strategic 
factors is noted until the contribution exceeds $50.00, when labor type 2 
is added to the group. Machine hours type 2 are added to the basis 
replacing ?/2, labor type 3 replaces labor type 1 allocated to product 1, and 
slack in labor type 2 replaces y^ at the points of discontinuity indicated 
above. Information of this kind is, of course, invaluable to the supervisor 
in determining the effect of ''errors" in the trading prices on production 
schedules. 

Treating the capacities of the strategic factors as parameters where 
their expansion is a possibility will also yield valuable information. For 



iFor parameterizations via efficient (Pareto) points see the method described 
by Charnes and Cooper in "Theory and Computation for Delegation Models: 
K-Efficiency, Functional Efficiency and Goals" (14). 



140 A Mathematical Model for Budgetary Planning 

example, it is advantageous to increase the availability of material 2 to 
533 units, provided the premium is not more than SO. 80 per unit; an 
increase of still another 7.35 units is justified provided the premium is not 
more than $0.30 per unit; no increase above 540.35 units is profitable 
unless one of the capacities of the other strategic factors is changed. 
Perhaps the most valuable aspect of the parametric programming is the 
preparation of a schedule of priorities so that the department head can, 
as his time permits, investigate, or cause to be investigated, the extension 
of those restrictions which seem to produce the most severe limitations 
on his activity. 

5.10. Extension of the Model to the Next Higher Level in the 
Hierarchy 

The outputs of the department, yi, constitute a set of factors seen by 
the supervisor of several departments. Together with the outputs of 
other departments, they form the raw materials for the outputs of the 
multi-department unit. Although it may seem at first circular to call the 
contributions of the original problem the costs for the multi-department 
unit, the ability of the supervisor to investigate the possibility of 
obtaining additional input by paying a premium (i.e., raising the 
contribution per unit output in the original problem) provides a frame- 
work for intercommunication between the levels without the supervisor 
actually investigating the behavior of the factors in the original problem. 
Furthermore he can, for short-run problems, forcibly raise the require- 
ments of certain outputs of the departments, obtaining from them the 
cost of so doing (from the Rik^ of the original problem). 

It is quite conceivable that a compounding of such models could be 
developed so that the outputs at each level could serve as the inputs to 
the next, etc., in such a way that an optimum optimorum could be found. 
This development, however, is beyond the scope of this thesis. Never- 
theless, the work of Charnes, Cooper, and Miller (18) toward developing 
an algorithm for the construction of algorithms strongly suggests that 
it may be possible to devise models at each level which would be appro- 
priate to the organization at each level. Pending the development of 
such an algorithm, the hierarchical formulation ^ rests on existing models 
which, in the main (the model of this thesis excepted) were not developed 
for the primary purpose of coordinating the activities of groups. 



^Not to be confused with the "hierarchical model" of linear programming — vide 
Cooper and Charnes (10), or of experimental design — vide Kempthorne(4:l), 



A Mathematical Model for Budgetary Planning 14-1 

5.11. Summary 

It should be emphasized once again that the above model was devised 
for studying the problems of planning and coordination. In particular, 
it has been an attempt at devising a scheme for coordinating control 
efforts in a multi-department situation. Developed specifically for the 
purpose of studying the interrelationships of factors in logical grouping 
rather than individually, it lends itself naturally to the study of more or 
less separated collections of activities in a firm. Through its relatively 
simple computation techniques, it is possible to assess the effect on 
output of variations in the input of any of the factors in any of the 
groups. Furthermore, it is possible to study the effect of the availability 
of a factor in one of the groups on the usage in another. 

Thus the model provides a vehicle for the study of substitutability 
and complementarity of factors in a setting which more closely resembles 
the operation of an actual firm than the classical economic model of the 
firm. The increase in availability of one of the factors may lead to an 
increased usage of another in one of the other groups of factors — i.e., 
''factor cooperancy" — but may cause no change in some of the others. 
Or, an increase in availability in one of the factors in a subgroup may 
exhibit "factor rivalry," causing a decrease in use of another, although 
(at least within certain limits) it may not. In either case the model offers 
a convenient means of studying these relationships. 

Returning to issues which are more closely related to this thesis, its 
relevance to specific types of planning should be noted. From the 
standpoint of planning for capital expenditures, the model (provided 
the costs and contributions are accurate) is an extremely convenient 
device for determining total cost or profit under various equipment plans, 
e.g., machine hour capacities. Similarly, planned expansion or contrac- 
tion in various segments of the labor force may be tested with the aid 
of the model. 1 

Utilizing an essentially Marshallian (60) concept, the effect of prices 
on output can be studied under assumptions which do not require infinite 
substitutability of all production factors. Furthermore, the effect of 
changes in output "prices" on the prices which could (or should) be paid 
to increase the availability of factors can be studied through the dual 
variables of the linear programming formulation. Having implications 
both with regard to planning and control, the effectiveness of using 



^This process of varying the capacity restrictions — i.e., parametric programming 
in the dual — is essentially equivalent to the systematic varying of costs or contri- 
butions in the direct problem. 



142 A Mathematical Model for Budgetary Planning 

intra-company trading prices as a control de\dce (with rate of profit 
increase or a profit goal as a criterion of efficiency) can be studied along 
with the necessary coordination of inputs of one "control unit" with 
the outputs of another under such a scheme. 

Some further inference may be drawn with respect to control ^-iewed 
in the light of one individual controlling several interdependent factors 
(or departments) rather than as the single factor control dealt with 
elsewhere in this paper. First of all, the dual variables give "clues" as to 
where it is important to attempt to carefully control certain factors. 
In the context presented by Charnes and Cooper (11), the model can be 
used to at least determine some priorities in the allocation of super^'isors' 
time. 

Also, the possibility of actually preventing an increase in profit 
through certain types of control is apparent by the use of the model. 
In particular, "management by exception" which deals primarily with 
unfavorable deviations from standard may produce a suboptimization 
in terms of the price or usage of one or more factors which does not lead 
to optimization in the large. A possible impUcation is that both positive 
and negative (favorable and unfavorable) variances must be examined 
in an effective control system. 

The development of a hierarchj^ of such (or similar) models would be 
very desirable for the extension of the power of this one-department 
model to the multi-level organization met in practice. Pending such 
development, however, the implications of the model for coordination 
at any level may be used, with reservations, of course, implied by the 
dangers of local optima. 

To carry this analysis somewhat further, it is to be noted that 
suboptimization within any subgroup \\dll not produce an overall 
optimum unless the output stipulations of the overall optimum are 
restrictions on the subproblems. Furthermore, examination of the 
optimal tableau indicates that in some instances, the cheapest means 
of producing product equivalents is not always used. That is, a product 
equivalent of two factors may be produced most cheaply by the use of 
the same factor. However, the relative advantage of one may be far 
greater than for the other, and hence this former may well utilize all or 
most of this cheapest factor. If the cost of each product is separately 
controlled, it may lead to some compromise in order to prevent negative 
variances in one of the costs. The deleterious effect of such mdividual 
product control of costs within the subgroup would be accentuated if 
the standard for the product whose relative advantage was greater was 
"loose," the other "tight." 



A Mathematical Model for Budgetary Planning 143 

Thus the criteria of "accurate costs" and ''control exercised on an 
individual product basis in every department where possible" applied 
to standards are shown, through use of the model, to produce nonoptimal 
results. In the case described, the standard for the product whose 
relative advantage for the use of the shared factor is greater should be 
"tight" and the other "loose," ceteris paribus, to produce an optimal 
result. Furthermore, the criteria for the setting of individual standards 
are not independent of the output requirements and hence are not 
independent of the activities (and standards) of the other subgroups. No 
amount of time and effort spent on the setting of "accurate" engineering 
standards for an individual activity can eliminate this interdependence. 

The model thus points out the need for a thorough re-evaluation of 
the criteria of setting of budgets and standards. It suggests the study of 
the planned use of incorrect information in setting individual standards 
which would lead to an optimum in the large. 



CHAPTER 6 



Summary and Directions for 
Further Research 



6.1. Introduction 

The preceding chapter concludes the substantive and methodological 
presentations for this thesis. This chapter will attempt to (1) draw 
together in summary fashion some of the main points and conclusions, 
(2) establish some degree of relevance to the existing literature in 
budgetary (and related) practice, and (3) indicate possible directions 
which appear promising for further research. In passing through this 
order of topics it will also be possible to point up (still further) certain 
shortcomings and gaps in the present study which are pertinent to 
possible attempts at actual application. 

The main body of the thesis may be divided as follows: 

1. An introductory statement of the problem of budgetary control, 
its setting, and the motivation underlying this study (Chapter 1). 

2. An analytical model to establish certain broad qualitative features 
of the topic to be studied in more precise form (Chapter 2). 

3. A survey (and assessment) of relevant bodies of scientific hypoth- 
eses and supporting experimental evidence and, to a lesser extent, 
managerial experience which could be used to point up issues both 
of theory and date inadequacy (Chapter 3). 

4. A report of an experiment undertaken to test certain major parts 
of the theory presented as well as collateral issues which are 
relevant to psychology, economics, and management practice 
(Chapter 4). 

144 



Summary and Directions for Further Research 145 

5. Finally, a new analytical model along with a method of solution 
and analysis designed to (a) highlight certain features, such as 
interdepartmental effects considered over a hierarchy, which were 
not specifically covered in the preceding materials and to (b) 
state preliminary propositions which appear plausible enough to 
warrant further investigation ; to do so, moreover, in a way which 
is not so ambiguous as to escape any possibility of scientific verifi- 
cation or rejection by recourse to suitably controlled situations. ^ 

6.2. Planning and Control 

At this stage of the analysis it is well to note certain differences 
between the approach used in Chapter 5 and elsewhere in this thesis. 
Whereas most of the emphasis has been on controls applied to individual 
behavior, the model of Chapter 5 is oriented as well towards interactions 
and coordination of a purely planning nature. ^ The latter is thus more 
closely akin to the kind of approach (alternative costs considered mutatis 
mutandis) which has been typical in economic analysis and which has 
been carried over into operations research. It is based on a consideration 
of alternatives as if the alternative chosen will in fact be carried out to 
a reasonable degree of approximation. This theory is best viewed in 
terms of an individual decision-making model where (assuming adequate 
physiological responses) the organism may be expected to react faithfully 
and exactly to its own commands. In such a system^ there is no real 
problem of goal discrepancies (e.g., of a budget vs. aspiration level 
variety), and virtually everything reduces to the question of whether 
accuracy and felicity are present in the costs and benefits that have been 



iJt is believed that the model and methods adduced in Chapter 5 make some 
contribution to the planning literature (and methodology) as found in management 
science, operations research, etc. — especially to that part of the literature (and 
research) which is concerned with "model types." On the other hand, this is not the 
main objective here since, unlike the typical applications in management science, 
the purpose is ultimately to devise certain explicit tests that involve motivation and 
control elements. Thus, for this purpose, it is hoped ultimately to devise a suitable 
experiment which would test the consequences of supplying programming results of 
an optimizing variety to subordinates, as against other kinds of information, in order 
to ascertain when, and under what circumstances, one or the other of these kinds of 
information might be preferable. 

^Cf. Charnes and Cooper, (10) for a further discussion of the difference between 
"planning" and "control," and Chapter 1 of this thesis. 

^At least with the assumptions which are usually used in the analyses. 



146 Summary and Directions for Further Research 

calculated.^ It follows that there also is no problem which can result 
from the effects of goal discrepancies upon performance. 

In contrast to this ''economic-planning" approach — which at first 
seems to be of a multi-person variety but is, in reality, a single-person 
multiple-factor approach — even the single-person model of this paper 
is seen to be a multiple-person approach. (Otherwise goal discrepancies 
could not be maintained.) The issue of "rational" calculation is then 
seen to assume dimensions which are different from those which might 
be expected to apply to planning considerations only, i.e., interpersonal 
dimensions. For control (as distinct from planning) purposes the 
utilization of accurate calculations and the transmission of their results 
to others may be beneficial, neutral, or harmful depending on (a) the 
kinds of persons and tasks involved, (b) the setting in which operations 
are conducted, and (c) the vehicle and the time sequences, etc., utihzed 
for information transmittal. 

A servomechanical analogy may be helpful. ^ As is well kno^^^l, the 
accuracy of calibrations and the speed of response t changes in the 
system (or the environment) that may, or should, be incorporated in 
such a mechanism cannot be decided apart from other characteristics 
of the system. It is possible, for example, that more "accurate" or more 
timely — or even more timely and accurate — "feedback" of information 
may fail to improve performance and may even have harmful effects. 
While it is not proposed to push this analogy with servomechanisms to 
absurd hmits, it seemed plausible to suppose that systems involving 
humans, or humans and machines, might be subject to the same general 
kinds of considerations. Thus, an hypothesis which is part of certain 
servomechanism designs, deals with issues involving rates of change 
(rather than levels) imposed on certain control elements and the effects 
that such changes may have on performance. The experimental evidence 
is wholly compatible with this hypothesis and stands in rather sharp 
contrast to much of the literature of budgeting (and economics), which is 
preoccupied with "levels" and the accuracy thereof. 

It should be said, however, that most of the situations envisioned in 



iln fact, a good deal of the literature in economics is concerned with formulating 
methods — graphs, calculus, linear programming, etc. — to facilitate such calcula- 
tions and/or prescribing certain necessary or sufficient conditions which may be used 
to verify whether the wanted results will be achieved. 

2If a justification for this analogy is desired, it may be secured from the comments 
of W. W. Cooper and H. A. Simon. See their commentary on the paper by F. Modig- 
liani and H. Sauerlender in Short Term Economic Forecasting (42a, b), pp. 352 ff. 



Summary and Directions for Further Research 147 

the industrial literature do not have learning components involving the 
same orders of magnitude for rates of learning that were present in the 
experiment.^ On the other hand, there is no reason to assume that all 
operations are conducted along the frontier provided by an optimal 
production function, as is assumed in economics. Nor is there reason to 
assume that the general prescription of budgets (''not-too-loose, but 
attainable") which is virtually ubiquitous in the management literature 
is always "best" — or even generally "best."^ 

While rules for change are given in the management literature on 
budgeting, these are generally assumed to be required only when struc- 
tural changes (e.g., due to technological alterations) occur, provided the 
budget level was "accurate" in the first place. In particular, the idea 
of using change (at suitable rates and upwards as well as downwards) as 
a control instrument per se is not systematically discussed and, for 
the most part, is not even mentioned.^ Both the experiment and the 
analytical model which preceded it indicate circumstances in which 
"levels," howevtD: "accurate," will not induce performance which is both 
possible and desirable. On the other hand, when rates of change are to be 
used then more careful analyses (and more powerful techniques) are 
required since, under certain circumstances, their use may have explosive 
effects with highly deleterious consequences."* 

However the particular details of the experiment and model may 
be viewed, it seems at least reasonable to suppose that it is a proper 
task of budgetary control to be concerned with strategies for constant 
improvement of performance. This cannot be done, of course, and it 
even reduces to nonsense if the presuppositions of economic theory 
are adopted. No explicit test has been made of the hypothesis that 



^But see footnote 1, p. 67, supra. Although the rates of learning might not be 
comparable, according to the definition of task difficulty defined by Chapman, 
Kennedy, Newell, and Biel (8), the rate of change of difficulty of the experimental 
task could be approximated in the industrial task. 

2As has been repeatedly noted, the Uterature of budgeting (and industrial engi- 
neering) and the literature of economics are not wholly consistent, relative to each 
other, in their assumptions or even in their use of the term "best." 

^This may possibly be the result of reactions to the hostile criticisms of labor 
(and others) to early industrial engineering practice, which is supposed to have 
utilized directed incentives for unilateral and unidirectional changes in worker 
performance requirements. It seems equally plausible to believe, however, that is is 
rather a byproduct of the emphasis on "accuracy." 

^This is well documented in the servomechanical literature and is explicitly 
considered (in a somewhat different fashion) in postulate (iiic) of the analytical model 
discussed in Chapter 2. 



148 Summary and Directions for Further Research 

individual firms do, in fact, operate on an optimum production function 
although, as pointed out in Chapter 1, the studies of Cyert, Dill, and 
March (23) and Lanzillotti (47) offer evidence to the contrar>^ For 
whatever it is worth — and it is not offered as valid scientific evidence — 
the reference to standards that are "not-too-loose, but attainable," which 
appears in many parts of the management Hterature, may be offered as 
an example that the assumed situation does not obtain ubiquitously. 
It is difficult, moreover, to believe that most plants are actually so well 
run and so well organized that a supervisor cannot effect some further 
improvements even though the state of technology is already well fLxed. 
In addition, it is not always clear just what is meant by the technological 
optimum. Consider, for instance, a student who fails to make a perfect 
score on an examination. The technical knowledge being available (at 
least to the instructor), does this mean that the student fails to achieve 
an optimum even though his performance was brilliant? Or, to address 
both the management and economics literature, does a fundamental 
(technological) change take place whenever a virtuoso renders a new 
and better interpretation of a known piece of music? A chess or bridge 
player may, through practice, learn to improve his game even though 
no fundamental change in rules is involved. To come closer to the 
problems of the individual firm, it has been observed that typists and 
secretaries improve their performance over repeated trials ^^-ith their 
machines and/or their bosses. 

It is not proposed to be unduly critical in these observations or even 
to overdraw the distinctions between planning and control. In practical 
situations the two (e.g., the budget as a plan and as a control) may well 
be confounded and, as already noted, ^ even the planning model discussed 
in Chapter 5 may be extended for control studies. Consider, for example, 
the light it might throw on the calculation (and use) of ''accurate" 
(levels) of standards and the ''principle of exceptions." One version 
of the principle of exceptions involves symmetric reporting of both 
favorable and unfavorable "variances. "^ Another involves reporting 
(and presumably investigation) of unfavorable variances onl^^ The 
presumption in the first case is that knowledge of causes is desirable 
for future improvement as well as immediate correction. The second 
focuses on the latter only, perhaps depending on other devices such as 



^Cf., supra, p. 3 ff. and p. 141. 

2This term is used in its accounting rather than its statistical sense. See, however, 
R. M. Cyert and G. Meyers (80a) for an application of statistical principles as a guide 
to variance reporting, as well as a suggestion as to how these techniques may be used 
to determine relatively tight and loose standards. 



Summary and Directions for Further Research 149 

informal observation or, possibly, periodic investigation to determine 
whether structural changes sufficient to warrant a change in standard have 
occurred. In either case it is generally presumed, perhaps according to 
some ordering, that an expenditure of managerial time and attention is 
warranted and that the effects will be beneficial — or at least not baneful. 

This topic, conditions for reporting variances and (higher level) 
managerial invention, can (and it is hoped will) be investigated. But 
even before this is done the model of Chapter 5 demands that certain 
other issues related to accurate standards and variance reports (principle 
of exceptions) be attended to. In particular, the parameterization 
techniques revealed certain cases where a premium (above standard) 
was warranted. This means that if the variance were eliminated and 
the standard were reduced to the level where it faithfully reflected the 
true performance possibility of an individual department, then if the 
investigation were successful in bringing performance into line, an overall 
worsening would occur. Thus, at the cost of managerial time and effort, 
the profits of the entity would be reduced. 

It does not follow, of course, that a report of the true dual values to 
the supervisor of this department would necessarily produce the desired 
result either. All that can be said at this juncture is that the topic is one 
which warrants serious investigation, and this is one among other issues 
which the model of Chapter 5 has helped to uncover — and in a form 
which may be readily extended for future testing. 

6.3. Some Selected Quotations 

At this point it may be well to introduce selected quotations from 
the management literature which will help to supply perspective on the 
preceding comments and to prepare the way for the concluding portion 
of this chapter, which deals with some limitations of the present study 
and indicates some possible future directions for research. It has not 
been possible to draw upon a representative sample of this literature, 
so the author's choice and possible biases should be allowed for. On the 
other hand, an effort was made to be fair, and quotations were selected 
w^hich would not merely document the previous comments but would 
also serve to indicate some of the qualifications that writers in the field 
typically employ in their presentations.^ 



^As a result some of the statements may appear unnecessarily vague. This seems, 
however, to be an unavoidable consequence of eliminating from inclusion in this series 
statements of a more clear and succinct character which, out of context, might appear 
merely as an attempt at setting up a series of "straw men." 



150 Summary and Directions for Further Research 

Consider, first, the following statements from Rautenstrauch and 
Villers (64), which give their views on budgeting for fixed and varia- 
ble expense: 

The fixed expense which does not vary with production is budgeted 
as for the previous period, unless some fundamental changes have been 
made involving fixed charges. If, for instance, some new equipment 
has been bought, the depreciation expense is increased. The budgeting 
of the fixed expense does not as a rule encounter any serious difficulty. 

The variable expense — The same can be said of the variable compo- 
nent of the factory overhead (such as indirect material) . Past records 
will generally provide the necessary data. As the variable expense varies 
directly with production, the data will easily be recomputed in terms 
of the future rate of production, according to the production budget. ^ 

These all-purpose statements are intended to cover in somewhat 
mechanical fashion all sorts of ''human" and ''business" situations. 
The budget pattern of the experiment which best approximates the 
procedures outlined is the "low" budget. The logically derived result 
of a static budget in the mathematical model is a static cost. In the 
standard procedures described there is no incentive for the search for 
and alteration of strategic factors described in Chapter 5. 

The following questions might now be asked: Is it necessarily true 
that "fundamental changes" must be made (even assuming an "accu- 
rate" budget) before scheduling an improvement in performance? What 
is said of the reverse problem of when (e.g., via search stimulation) 
a request for authorizing new equipment should take place at the 
departmental level? In addition, what incentive (and risks) should be 
provided for this purpose? Clearly there is a possible interaction between 
fixed and variable expense that may be possible in many conceivable 
situations. There is a further need for considering possible interdepart- 
mental reactions of the kind discussed in Chapter 5, and this is far from 
being a simple problem (with past records providing the necessarj^ data) ; 
it involves, as a matter of reporting and budgeting design, very subtle 
problems in economics and psychology. How, in general, is the pos- 
sibility of continued improvement in performance to be handled — or is 
this problem irrelevant as an issue of budgeting? 

The above quotations are from a standard text in industrial engi- 
neering. ^ One of the better books on budgeting presented from an 



1(64), pp. 132-133. 

2Cf. the short biographical note on Dr. Villers (and his association with the late 
Dr. Rautenstrauch) which appears opposite his article, "Industrial Budgeting," 
pp. 1073 ff., in W. G. Ireson and E. L. Grant (38). 



Summary and Directions for Further Research 151 

accounting standpoint is J. Brooks Heckert's Business Budgeting and 
Control (34). It may be useful, therefore, to turn to this source for a 
quotation on another of the topics mentioned in this thesis. After making 
a clear (and cogent) distinction between standards (for judging perform- 
ance) and budgets (for planning purposes), ^ Heckert goes on to say: 

In many concerns where operations have become highly stand- 
ardized, the distinction (i.e., between budget and standard costs) tends 
to disappear and the budget figures serve, generally speaking, both as 
measures of performance and as a coordinating tool. This is particularly 
true in regard to production operations and costs and the more mechan- 
ical aspects of distribution activities; however, the distinction seldom 
disappears entirely. ^ 

It is not clear whether Heckert regards this tendency to coalescence 
as good or bad. More perceptive than many authors, he also tends to be 
more careful and guarded in his presentations. Within the limits of a 
possible interpretation, however, the following questions might be raised. 
Insofar as the two sets of external goals on the same data (e.g., costs) fail 
to coalesce, how much of a discrepancy should be tolerated? Is there 
some degree of discrepancy which constitutes a (psychological or logical) 
contradiction either in a department or over the entire system? If a 
contradiction is caused, what are its consequences for performance?^ 
If coalescence is to be encouraged, at what rate should it be introduced? 
More generally, what relations should such external goals bear (a) to 
internal goals, such as aspiration levels, and (b) to performance? 

In contrast to the extremely mechanical emphasis of most of the 
budgetary literature, the following statement by Heckert on sales quotas 
(and their relations to standards) is of interest: 

^ Actual experience with sales quotas, as with all standards, will reveal 
that sales representatives react to them somewhat differently, partic- 
ularly at first. Some are stimulated to their highest efficiency, while 



^The distinction drawn by Heckert is related to, but not identical with, the one 
previously made in this chapter — i.e., the distinction between the budget as a plan 
and as a control instrument. Actually, the distinction used in this thesis is developed 
in an unpublished manuscript by W. W. Cooper (21), "Historical Cost and Alter- 
native Cost," to which I was given access. 

2(34), p. 10. 

3Cf. Charnes and Cooper, "Silhouette Functions of Cost Behavior" (12), Quarterly 
Journal of Economics, for a discussion of possible contradictions (in a logical sense) 
and also their discussion "On The Theory and Computation of Delegation Models" 
(14) for a discussion of the possible use of contradictions in managerial planning along 
with a specification of analytical methods for dealing with these problems in a 
mathematical context. 



152 Summary and Directions for Further Research 

others are discouraged. Some sales executives place considerable 
emphasis upon this human element in setting their quotas. In general, 
however, good men will, in the long run, respond favorably to intel- 
ligently devised quotas, particularly when compensation is fairly 
adjusted to performance.^ '' 

Although the meaning of ''intelligently devised quotas" is not 
amplified, the above remarks tend to imply that "some sales executives" 
probably have a better plan for improving performance through bud- 
geting than those described in the traditional books on budgeting. 
Certain fairly obvious questions may be raised about parts of the 
statements as quoted, but it is perhaps best to let the matter rest here in 
order to turn attention to some other topics which are at least indirectly 
relevant to what Heckert seems to prescribe. 

Apt statements may be found almost at will on the principle of excep- 
tions. It is therefore of interest to indicate that, under the conditions 
over which it was conducted, the experiment indicates that variations 
from standard are neither an accurate measure of performance nor an 
accurate guide to improvement in performance. Thus budget attainment 
in the ''low" budget ^ groups was relatively high but performance was 
relatively low. According to these experimental results, the difficulty 
of achievement must either be arrived at by the frequency of attainment 
(i.e., a statistical analysis of past data) or by some independent measure 
(e.g., a problem-solving equivalent of Methods Time Measurement). 
This type of measurement would be referred to in practice as an "engi- 
neering standard," and to this extent the findings may tend to justify 
some of the distinctions and emphases found in the literature. 

On the other hand, the experiment offers some evidence for limita- 
tions on the value of "engineering standards." At the beginning of the 
experiment the performance capabilities on this task were almost 
unknown to the experimenter. Had 10,800 subjects been tested under a 
situation in which, for example, no rewards were present, the increment 
in information apphcable to the choice between a high and a low budget 
would have been insignificant. The results indicate that performance in 
a problem-solving task is really not determinate except under conditions 
in which the goals and rewards are specified. 

To draw some summary conclusions on this literature the follo^^dng 
observations are offered. Two important features seem to be lacking if 



1(34), p. 138. 

^Recall that this is the one which corresponds to the case usually recommended, 
"not-too-loose, but attainable." 



Summary and Directions for Further Research 153 

budgetary practice is to be judged by this literature. The first is a lack 
of a systematic empirical approach or of any systematic research that 
can serve as a basis for generalizations, evaluation, and further progress. 
If standards are to be set on the basis of historical cost and/or engi- 
neering standards, the possibihty of attaining better performance by the 
simple means of expecting it (or at least convincing the department 
head that better performance is expected) is a phenomenon which, if 
investigated, is not reported in the extant literature. 

The maintenance of a static budget and its possible failure to induce 
improvement in performance when that is possible has already been 
commented upon. The effects of rate changes and the conditions or 
amounts in which they might be applied are, by and large, ignored. To 
turn the usual point of view around, one wonders what would happen if 
the budget department were scored by red and black variances by 
reference to a technological optimum or ideal standard (if one could be 
constructed) of possible budgetary practice. 

The second of the two observations to be made is the curious phenom- 
enon of a tendency toward persistent disregard for the motivation 
structure, even at the level of the individual manager, despite the 
preoccupation of many other parts of managerial literature with this 
kind of problem. Emphasis is on ''accuracy" of the budget, not its 
relationship to the desires, capabilities, or varying situations in which a 
person who is held to such a budget may possess when he receives it. Ex- 
ceptions may be cited (as in the above quotation drawn from Heckert). 
Having said this much, however, it should probably also be said that 
many of the written rules are tempered in practice ; some may be wholly 
replaced and rules and procedures may also be employed that are not 
reported in the standard literature. This in itself, however, would be 
unfortunate and might be counted as a "lack." Perhaps a greater and 
more extended use of analytical models might have further use than as 
a guide to laboratory experiments, qualitative characterizations, or 
statistical-mathematical analyses. The explicit and exact statement that 
such models require might serve as a standard of reference against which 
new procedures or significant deviations in old procedures might be 
detected and judged. If this could be accompanied by publication then 
a foundation for rapid progress in business practice and scientific under- 
standing in this important area of management might be secured.^ 



iThe reference is to "control." Numerous mathematical models and modes of 
analyses for various phases of "planning" — e.g., for ascertaining or imputing the 
cost of funds — are, of course, already available. 



154 Summary and Directions for Further Research 

6.4. Some Directions for Further Research 

It has probably not escaped the reader's attention that I have 
perhaps been more harsh than can be justified in my statements about 
the budgeting hterature. If this is true then some ameUoration is due 
these authors. It can perhaps be attained by emphasizing, at the close 
of this thesis, certain highly important topics which have not been 
covered in either the analytical models or experimental results. This 
will be done in a form which suggests some further avenues of research. 

1. No attempt was made to analyze either timeliness or content of 
budget or accounting reports. Is understandability (hence 
simplicity or directness) a desideratum, as is often assumed? 
If so, are uniform reports to all individuals a proper medium for 
attaining this goal? Is there a dynamics (e.g., of learning) which 
should be allowed for? 

2. Only a short span of time was covered in the experiments. Would 
the effects uncovered hold true over longer spans, or would some 
of them weaken and others arise to take their place? In terms of 
the possible correctives suggested (or implied) at various points 
in the presentation of this thesis, the following quotation from 
P. A. Samuelson is particularly apt: "[I] . . . wish to point out 
the possible occurrence in economic systems of the common 
medical phenomena whereby short term remedies may have long 
term deleterious effects."^ This kind of possibility was not even 
considered in its own right in the analytical models presented. 

3. The question of intervention rules for superiors in the hierarchical 
model was not explored, so this thesis, like some of the writers 
quoted, is subject to criticism on this score. 

4. The relation of the internal environment of the individual to that 
of the firm and the relation of the latter (or, better, both) to the 
broader setting of society was not examined. What effects may 
be expected from this quarter on (a) aspiration level and (b) 
performance by both superiors and subordinates? 

Many other questions and deficiencies might be noted but enough, by 
now, has probably been said. To close, it is well to emphasize once more 
that this thesis has been submitted only as part of a scientific inquiry in 
an important area of management. If it arouses some interest even in 
criticizing, repairing, or extending some of the deficiencies of this study, 
it will have served its purpose. 



1(69), p. 355 (In 1947 ed.).In spite of the obvious veracity of this statement and its 
often unrecognized appHcabihty, it has apparently been omitted from the later edition. 



BIBLIOGRAPHY* 



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*This listing does not claim to be a complete bibliography of budget control 
or even a complete listing of those works consulted during the preparation of the 
thesis. It contains only those works which were specifically referred to in the text 
and footnotes. 

155 



156 Bibliography 

(13) Charnes, A., and W. W. Cooper, "The Stepping Stone Method of Ex- 
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(15) Charnes, A., W. W. Cooper, and A. Henderson, An Introduction to Linear 
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(17) Charnes, A., W. W. Cooper, and B. Mellon, "A Model for Optimization 
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July, 1955. 

(18) Charnes, A., W. W. Cooper, and M. H. Miller, "Dyadic Problems and 
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nology, Graduate School of Industrial Administration, Office of Naval 
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(19) Clark, J. M., Studies in the Economics of Overhead Costs, Chicago: The 
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(20) Cooper, W. W., "A Proposal for Extending The Theory of The Firm," 
Quarterly Journal of Economics, February, 1951. 

(21) Cooper, W. W., "Historical and Alternative Costs: A Study of Some 
Relations Between the Economic Theory of the Firm and the Accounting 
Control of Operations," Unpublished Doctoral Dissertation, Columbia 
University, 1950. 

(22) Crandall, V. J., "A Preliminary Investigation of the Generalization of 
Experimentally Induced Frustration in Fantasy Production," Unpublished 
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Rotter (67). 

(23) Cyert, R. M., W. R. Dill, and J. G. March, "The Role of Expectations in 
Business Decision Making," Administrative Science Quarterly, Vol. 3, No. 3, 
December, 1958. 



Bibliography 157 

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(26) Dorfman, R., P. A. Samuelson, and R. M. Solow, Linear Programming and 
Economic Analysis, New York: McGraw-Hill Book Company, Inc., 1958. 

(27) Dreze, J. H., ''Individual Decision Making Under Partially Controllable 
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1958. 

(28) Drucker, P., The Concept of The Corporation, New York: John Day and 
Company, 1946. 

(29) Edwards, W., "The Prediction of Decisions Among Bets," Journal of 
Experimental Psychology, 1955, pp. 201-214. 

(30) Edwards, W., "Probability-Preference in Gambling," American Journal of 
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(31) Edwards, W., "The Theory of Decision Making," Psychological Bulletin, 
Vol. 51, 1954, pp. 380-417. 

(32) Fisher, R. A., "Statistical Methods and Scientific Induction," Journal of 
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(33) Frank, J. D., "Individual Differences in Certain Aspects of the Level of 
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\34) Heckert, J. B., Business Budgeting and Control, New York: The Ronald 
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(38) Ireson, W. G., and E. L. Grant (eds.), Handbook of Industrial Engineering 
and Management, Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1955. 



158 Bibliography 

(39) Jessor, R., "A Methodological Investigation of the Strength and General- 
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Ohio State University, 1951, cited in J. B. Rotter (67). 

(40) Keller, I. W., Management Accounting for Profit Control, New York: 
McGraw-Hill Book Company, Inc., 1957. 

(41) Kempthorne, 0., The Design and Analysis of Experiments, New York: 
John Wiley and Sons, Inc., 1952. 

(42) Klein, L. R. (ed.), Short Term Economic Forecasting, Vol. 17 in Studies in 
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(43) Knight, F. H., Risk, Uncertainty, and Profit, London: London School of 
Economics and Political Science Series of Reprints of Scarce Tracts in 
Economics and Political Science, 1946. 

(44) Kohler, E. L., A Dictionary for Accountants, Englewood Cliffs, N. J.: 
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(45) Koopmans, T. C. (ed.). Activity Analysis of Production and Allocation, 
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and A. W. Tucker, "Linear Programming and the Theory of Games;" 
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of Activities." 

(46) Lang, T., W. B. McFarland, and M. Schiff, Cost Accounting, New York: 
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(47) Lanzilotti, F., "Pricing Objectives in Large Compsimes," American Economic 
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(48) Lasser, J. K. (ed.). Handbook of Cost Accounting Methods, New York: 
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(49) Leavitt, H. J., Managerial Psychology: An Introduction to Individuals, 
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(50) Lincoln, J. F., Incentive Management, Cleveland: The Lincoln Electric 
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(53) MacDonald, J. H., Practical Budget Procedure, Englewood Cliffs, N. J.: 
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(54) MacGregor, D., "An Uneasy Look at Performance Appraisal," Harvard 
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(55) Mann, H. B., Analysis and Design of Experiments: Analysis of Variance and 
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(56) Manne, A. S., Scheduling of Petroleum Refinery Operations, Cambridge: 
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(57) March, J. G., and H. A. Simon, Organizations, New York: John Wiley and 
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(58) Marschak, J., "Elements for a Theory of Teams," Management Science, 
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(59) Marschak, J., "Rational Behavior, Uncertain Prospects and Measurable 

Utility," Econometrica, Vol. 18, No. 2, April, 1950. 

(60) Marshall, A., Principles of Economics, London: Macmillan and Company, 
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(61) McFarland, W. B., "The Basic Theory of Standard Costs," The Accounting 
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(62) Modigliani, F., and F. E. Hohn, "Planning Over Time, with Some Con- 
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(63) Mosteller, F., and P. Nogee, "An Experimental Measurement of Utility," 
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(63a)Radner, R., "The Apphcation of Linear Programming to Team Decision 
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(64) Rautenstrauch, W., and R. Villers, Budgetary Control, New York: Funk 
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(65) Robbins, H., "Some Aspects of the Sequential Design of Experiments," 
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(67) Rotter, J. B., Social Learning and Clinical Psychology, Englewood Cliffs, 
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160 Bibliography 

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